Algebra Examples

$9 x^{2} + 4 {(y - 2)}^{2} = 36$ $x^{2} = 2 y$

Step 1

Subtract $9 x^{2}$ from both sides of the equation.

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$x^{2} = 2 y$

Step 2

Replace all occurrences of $y$ with $36 - 9 x^{2}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $x^{2} = 2 y$ with $36 - 9 x^{2}$ .

$x^{2} = 2 (36 - 9 x^{2})$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $2 (36 - 9 x^{2})$ .

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Step 2.2.1.1

Apply the distributive property.

$x^{2} = 2 \cdot 36 + 2 (- 9 x^{2})$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 2.2.1.2

Multiply.

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Step 2.2.1.2.1

Multiply $2$ by $36$ .

$x^{2} = 72 + 2 (- 9 x^{2})$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 2.2.1.2.2

Multiply $- 9$ by $2$ .

$x^{2} = 72 - 18 x^{2}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$x^{2} = 72 - 18 x^{2}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$x^{2} = 72 - 18 x^{2}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$x^{2} = 72 - 18 x^{2}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$x^{2} = 72 - 18 x^{2}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3

Solve for $x$ in $x^{2} = 72 - 18 x^{2}$ .

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Step 3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.1.1

Add $18 x^{2}$ to both sides of the equation.

$x^{2} + 18 x^{2} = 72$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.1.2

Add $x^{2}$ and $18 x^{2}$ .

$19 x^{2} = 72$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$19 x^{2} = 72$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.2

Divide each term in $19 x^{2} = 72$ by $19$ and simplify.

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Step 3.2.1

Divide each term in $19 x^{2} = 72$ by $19$ .

$\frac{19 x^{2}}{19} = \frac{72}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $19$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{19 x^{2}}{19} = \frac{72}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{72}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$x^{2} = \frac{72}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$x^{2} = \frac{72}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$x^{2} = \frac{72}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{72}{19}}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4

Simplify $\pm \sqrt{\frac{72}{19}}$ .

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Step 3.4.1

Rewrite $\sqrt{\frac{72}{19}}$ as $\frac{\sqrt{72}}{\sqrt{19}}$ .

$x = \pm \frac{\sqrt{72}}{\sqrt{19}}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4.2

Simplify the numerator.

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Step 3.4.2.1

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 3.4.2.1.1

Factor $36$ out of $72$ .

$x = \pm \frac{\sqrt{36 (2)}}{\sqrt{19}}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4.2.1.2

Rewrite $36$ as $6^{2}$ .

$x = \pm \frac{\sqrt{6^{2} \cdot 2}}{\sqrt{19}}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$x = \pm \frac{\sqrt{6^{2} \cdot 2}}{\sqrt{19}}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4.2.2

Pull terms out from under the radical.

$x = \pm \frac{6 \sqrt{2}}{\sqrt{19}}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$x = \pm \frac{6 \sqrt{2}}{\sqrt{19}}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4.3

Multiply $\frac{6 \sqrt{2}}{\sqrt{19}}$ by $\frac{\sqrt{19}}{\sqrt{19}}$ .

$x = \pm \frac{6 \sqrt{2}}{\sqrt{19}} \cdot \frac{\sqrt{19}}{\sqrt{19}}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4.4

Combine and simplify the denominator.

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Step 3.4.4.1

Multiply $\frac{6 \sqrt{2}}{\sqrt{19}}$ by $\frac{\sqrt{19}}{\sqrt{19}}$ .

$x = \pm \frac{6 \sqrt{2} \sqrt{19}}{\sqrt{19} \sqrt{19}}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4.4.2

Raise $\sqrt{19}$ to the power of $1$ .

$x = \pm \frac{6 \sqrt{2} \sqrt{19}}{\sqrt{19} \sqrt{19}}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4.4.3

Raise $\sqrt{19}$ to the power of $1$ .

$x = \pm \frac{6 \sqrt{2} \sqrt{19}}{\sqrt{19} \sqrt{19}}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{6 \sqrt{2} \sqrt{19}}{{\sqrt{19}}^{1 + 1}}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{6 \sqrt{2} \sqrt{19}}{{\sqrt{19}}^{2}}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4.4.6

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

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Step 3.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$x = \pm \frac{6 \sqrt{2} \sqrt{19}}{{(19^{\frac{1}{2}})}^{2}}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{6 \sqrt{2} \sqrt{19}}{19^{\frac{1}{2} \cdot 2}}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{6 \sqrt{2} \sqrt{19}}{19^{\frac{2}{2}}}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4.4.6.4

Cancel the common factor of $2$ .

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Step 3.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{6 \sqrt{2} \sqrt{19}}{19^{\frac{2}{2}}}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4.4.6.4.2

Rewrite the expression.

$x = \pm \frac{6 \sqrt{2} \sqrt{19}}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$x = \pm \frac{6 \sqrt{2} \sqrt{19}}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4.4.6.5

Evaluate the exponent.

$x = \pm \frac{6 \sqrt{2} \sqrt{19}}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$x = \pm \frac{6 \sqrt{2} \sqrt{19}}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$x = \pm \frac{6 \sqrt{2} \sqrt{19}}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4.5

Simplify the numerator.

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Step 3.4.5.1

Combine using the product rule for radicals.

$x = \pm \frac{6 \sqrt{19 \cdot 2}}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.4.5.2

Multiply $19$ by $2$ .

$x = \pm \frac{6 \sqrt{38}}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$x = \pm \frac{6 \sqrt{38}}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$x = \pm \frac{6 \sqrt{38}}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{6 \sqrt{38}}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{6 \sqrt{38}}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{6 \sqrt{38}}{19}, - \frac{6 \sqrt{38}}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$x = \frac{6 \sqrt{38}}{19}, - \frac{6 \sqrt{38}}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

$x = \frac{6 \sqrt{38}}{19}, - \frac{6 \sqrt{38}}{19}$

$4 {(y - 2)}^{2} = 36 - 9 x^{2}$

Step 4

Solve the system $x = \frac{6 \sqrt{38}}{19}, 4 {(y - 2)}^{2} = 36 - 9 x^{2}$ .

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Step 4.1

Replace all occurrences of $x$ with $\frac{6 \sqrt{38}}{19}$ in each equation.

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Step 4.1.1

Replace all occurrences of $x$ in $4 {(y - 2)}^{2} = 36 - 9 x^{2}$ with $\frac{6 \sqrt{38}}{19}$ .

$4 {(y - 2)}^{2} = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2

Simplify $4 {(y - 2)}^{2} = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$ .

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Step 4.1.2.1

Simplify the left side.

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Step 4.1.2.1.1

Simplify $4 {(y - 2)}^{2}$ .

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Step 4.1.2.1.1.1

Rewrite ${(y - 2)}^{2}$ as $(y - 2) (y - 2)$ .

$4 ((y - 2) (y - 2)) = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.1.1.2

Expand $(y - 2) (y - 2)$ using the FOIL Method.

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Step 4.1.2.1.1.2.1

Apply the distributive property.

$4 (y (y - 2) - 2 (y - 2)) = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.1.1.2.2

Apply the distributive property.

$4 (y \cdot y + y \cdot - 2 - 2 (y - 2)) = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.1.1.2.3

Apply the distributive property.

$4 (y \cdot y + y \cdot - 2 - 2 y - 2 \cdot - 2) = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

$4 (y \cdot y + y \cdot - 2 - 2 y - 2 \cdot - 2) = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.1.1.3

Simplify and combine like terms.

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Step 4.1.2.1.1.3.1

Simplify each term.

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Step 4.1.2.1.1.3.1.1

Multiply $y$ by $y$ .

$4 (y^{2} + y \cdot - 2 - 2 y - 2 \cdot - 2) = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.1.1.3.1.2

Move $- 2$ to the left of $y$ .

$4 (y^{2} - 2 \cdot y - 2 y - 2 \cdot - 2) = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.1.1.3.1.3

Multiply $- 2$ by $- 2$ .

$4 (y^{2} - 2 y - 2 y + 4) = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

$4 (y^{2} - 2 y - 2 y + 4) = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.1.1.3.2

Subtract $2 y$ from $- 2 y$ .

$4 (y^{2} - 4 y + 4) = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

$4 (y^{2} - 4 y + 4) = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.1.1.4

Apply the distributive property.

$4 y^{2} + 4 (- 4 y) + 4 \cdot 4 = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.1.1.5

Simplify.

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Step 4.1.2.1.1.5.1

Multiply $- 4$ by $4$ .

$4 y^{2} - 16 y + 4 \cdot 4 = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.1.1.5.2

Multiply $4$ by $4$ .

$4 y^{2} - 16 y + 16 = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2

Simplify the right side.

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Step 4.1.2.2.1

Simplify $36 - 9 {(\frac{6 \sqrt{38}}{19})}^{2}$ .

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Step 4.1.2.2.1.1

Simplify each term.

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Step 4.1.2.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.2.2.1.1.1.1

Apply the product rule to $\frac{6 \sqrt{38}}{19}$ .

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{{(6 \sqrt{38})}^{2}}{19^{2}}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.1.1.2

Apply the product rule to $6 \sqrt{38}$ .

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{6^{2} {\sqrt{38}}^{2}}{19^{2}}$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{6^{2} {\sqrt{38}}^{2}}{19^{2}}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.1.2

Simplify the numerator.

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Step 4.1.2.2.1.1.2.1

Raise $6$ to the power of $2$ .

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 {\sqrt{38}}^{2}}{19^{2}}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.1.2.2

Rewrite ${\sqrt{38}}^{2}$ as $38$ .

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Step 4.1.2.2.1.1.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{38}$ as $38^{\frac{1}{2}}$ .

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 {(38^{\frac{1}{2}})}^{2}}{19^{2}}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.1.2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38^{\frac{1}{2} \cdot 2}}{19^{2}}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.1.2.2.3

Combine $\frac{1}{2}$ and $2$ .

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38^{\frac{2}{2}}}{19^{2}}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.1.2.2.4

Cancel the common factor of $2$ .

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Step 4.1.2.2.1.1.2.2.4.1

Cancel the common factor.

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38^{\frac{2}{2}}}{19^{2}}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.1.2.2.4.2

Rewrite the expression.

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38}{19^{2}}$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38}{19^{2}}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.1.2.2.5

Evaluate the exponent.

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38}{19^{2}}$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38}{19^{2}}$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38}{19^{2}}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.1.3

Raise $19$ to the power of $2$ .

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38}{361}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.1.4

Multiply $36$ by $38$ .

$4 y^{2} - 16 y + 16 = 36 - 9 (\frac{1368}{361})$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.1.5

Cancel the common factor of $1368$ and $361$ .

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Step 4.1.2.2.1.1.5.1

Factor $19$ out of $1368$ .

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{19 (72)}{361}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.1.5.2

Cancel the common factors.

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Step 4.1.2.2.1.1.5.2.1

Factor $19$ out of $361$ .

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{19 \cdot 72}{19 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.1.5.2.2

Cancel the common factor.

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{19 \cdot 72}{19 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.1.5.2.3

Rewrite the expression.

$4 y^{2} - 16 y + 16 = 36 - 9 (\frac{72}{19})$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 (\frac{72}{19})$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 (\frac{72}{19})$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.1.6

Multiply $- 9 (\frac{72}{19})$ .

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Step 4.1.2.2.1.1.6.1

Combine $- 9$ and $\frac{72}{19}$ .

$4 y^{2} - 16 y + 16 = 36 + \frac{- 9 \cdot 72}{19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.1.6.2

Multiply $- 9$ by $72$ .

$4 y^{2} - 16 y + 16 = 36 + \frac{- 648}{19}$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 + \frac{- 648}{19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.1.7

Move the negative in front of the fraction.

$4 y^{2} - 16 y + 16 = 36 - \frac{648}{19}$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - \frac{648}{19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.2

To write $36$ as a fraction with a common denominator, multiply by $\frac{19}{19}$ .

$4 y^{2} - 16 y + 16 = 36 \cdot \frac{19}{19} - \frac{648}{19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.3

Combine $36$ and $\frac{19}{19}$ .

$4 y^{2} - 16 y + 16 = \frac{36 \cdot 19}{19} - \frac{648}{19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.4

Combine the numerators over the common denominator.

$4 y^{2} - 16 y + 16 = \frac{36 \cdot 19 - 648}{19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.5

Simplify the numerator.

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Step 4.1.2.2.1.5.1

Multiply $36$ by $19$ .

$4 y^{2} - 16 y + 16 = \frac{684 - 648}{19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.1.2.2.1.5.2

Subtract $648$ from $684$ .

$4 y^{2} - 16 y + 16 = \frac{36}{19}$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = \frac{36}{19}$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = \frac{36}{19}$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = \frac{36}{19}$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = \frac{36}{19}$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = \frac{36}{19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2

Solve for $y$ in $4 y^{2} - 16 y + 16 = \frac{36}{19}$ .

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Step 4.2.1

Subtract $\frac{36}{19}$ from both sides of the equation.

$4 y^{2} - 16 y + 16 - \frac{36}{19} = 0$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.2

Simplify $4 y^{2} - 16 y + 16 - \frac{36}{19}$ .

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Step 4.2.2.1

To write $16$ as a fraction with a common denominator, multiply by $\frac{19}{19}$ .

$4 y^{2} - 16 y + 16 \cdot \frac{19}{19} - \frac{36}{19} = 0$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.2.2

Combine $16$ and $\frac{19}{19}$ .

$4 y^{2} - 16 y + \frac{16 \cdot 19}{19} - \frac{36}{19} = 0$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.2.3

Combine the numerators over the common denominator.

$4 y^{2} - 16 y + \frac{16 \cdot 19 - 36}{19} = 0$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.2.4

Simplify the numerator.

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Step 4.2.2.4.1

Multiply $16$ by $19$ .

$4 y^{2} - 16 y + \frac{304 - 36}{19} = 0$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.2.4.2

Subtract $36$ from $304$ .

$4 y^{2} - 16 y + \frac{268}{19} = 0$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + \frac{268}{19} = 0$

$x = \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + \frac{268}{19} = 0$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.3

Factor $4$ out of $4 y^{2} - 16 y + \frac{268}{19}$ .

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Step 4.2.3.1

Factor $4$ out of $4 y^{2}$ .

$4 (y^{2}) - 16 y + \frac{268}{19} = 0$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.3.2

Factor $4$ out of $- 16 y$ .

$4 (y^{2}) + 4 (- 4 y) + \frac{268}{19} = 0$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.3.3

Factor $4$ out of $\frac{268}{19}$ .

$4 y^{2} + 4 (- 4 y) + 4 (\frac{67}{19}) = 0$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.3.4

Factor $4$ out of $4 y^{2} + 4 (- 4 y)$ .

$4 (y^{2} - 4 y) + 4 (\frac{67}{19}) = 0$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.3.5

Factor $4$ out of $4 (y^{2} - 4 y) + 4 (\frac{67}{19})$ .

$4 (y^{2} - 4 y + \frac{67}{19}) = 0$

$x = \frac{6 \sqrt{38}}{19}$

$4 (y^{2} - 4 y + \frac{67}{19}) = 0$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.4

Divide each term in $4 (y^{2} - 4 y + \frac{67}{19}) = 0$ by $4$ and simplify.

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Step 4.2.4.1

Divide each term in $4 (y^{2} - 4 y + \frac{67}{19}) = 0$ by $4$ .

$\frac{4 (y^{2} - 4 y + \frac{67}{19})}{4} = \frac{0}{4}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.4.2

Simplify the left side.

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Step 4.2.4.2.1

Cancel the common factor of $4$ .

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Step 4.2.4.2.1.1

Cancel the common factor.

$\frac{4 (y^{2} - 4 y + \frac{67}{19})}{4} = \frac{0}{4}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.4.2.1.2

Divide $y^{2} - 4 y + \frac{67}{19}$ by $1$ .

$y^{2} - 4 y + \frac{67}{19} = \frac{0}{4}$

$x = \frac{6 \sqrt{38}}{19}$

$y^{2} - 4 y + \frac{67}{19} = \frac{0}{4}$

$x = \frac{6 \sqrt{38}}{19}$

$y^{2} - 4 y + \frac{67}{19} = \frac{0}{4}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.4.3

Simplify the right side.

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Step 4.2.4.3.1

Divide $0$ by $4$ .

$y^{2} - 4 y + \frac{67}{19} = 0$

$x = \frac{6 \sqrt{38}}{19}$

$y^{2} - 4 y + \frac{67}{19} = 0$

$x = \frac{6 \sqrt{38}}{19}$

$y^{2} - 4 y + \frac{67}{19} = 0$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.5

Multiply through by the least common denominator $19$ , then simplify.

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Step 4.2.5.1

Apply the distributive property.

$19 y^{2} + 19 (- 4 y) + 19 (\frac{67}{19}) = 0$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.5.2

Simplify.

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Step 4.2.5.2.1

Multiply $- 4$ by $19$ .

$19 y^{2} - 76 y + 19 (\frac{67}{19}) = 0$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.5.2.2

Cancel the common factor of $19$ .

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Step 4.2.5.2.2.1

Cancel the common factor.

$19 y^{2} - 76 y + 19 (\frac{67}{19}) = 0$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.5.2.2.2

Rewrite the expression.

$19 y^{2} - 76 y + 67 = 0$

$x = \frac{6 \sqrt{38}}{19}$

$19 y^{2} - 76 y + 67 = 0$

$x = \frac{6 \sqrt{38}}{19}$

$19 y^{2} - 76 y + 67 = 0$

$x = \frac{6 \sqrt{38}}{19}$

$19 y^{2} - 76 y + 67 = 0$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.6

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.7

Substitute the values $a = 19$ , $b = - 76$ , and $c = 67$ into the quadratic formula and solve for $y$ .

$\frac{76 \pm \sqrt{{(- 76)}^{2} - 4 \cdot (19 \cdot 67)}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.8

Simplify.

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Step 4.2.8.1

Simplify the numerator.

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Step 4.2.8.1.1

Raise $- 76$ to the power of $2$ .

$y = \frac{76 \pm \sqrt{5776 - 4 \cdot 19 \cdot 67}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.8.1.2

Multiply $- 4 \cdot 19 \cdot 67$ .

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Step 4.2.8.1.2.1

Multiply $- 4$ by $19$ .

$y = \frac{76 \pm \sqrt{5776 - 76 \cdot 67}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.8.1.2.2

Multiply $- 76$ by $67$ .

$y = \frac{76 \pm \sqrt{5776 - 5092}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm \sqrt{5776 - 5092}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.8.1.3

Subtract $5092$ from $5776$ .

$y = \frac{76 \pm \sqrt{684}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.8.1.4

Rewrite $684$ as $6^{2} \cdot 19$ .

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Step 4.2.8.1.4.1

Factor $36$ out of $684$ .

$y = \frac{76 \pm \sqrt{36 (19)}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.8.1.4.2

Rewrite $36$ as $6^{2}$ .

$y = \frac{76 \pm \sqrt{6^{2} \cdot 19}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm \sqrt{6^{2} \cdot 19}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.8.1.5

Pull terms out from under the radical.

$y = \frac{76 \pm 6 \sqrt{19}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm 6 \sqrt{19}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.8.2

Multiply $2$ by $19$ .

$y = \frac{76 \pm 6 \sqrt{19}}{38}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.8.3

Simplify $\frac{76 \pm 6 \sqrt{19}}{38}$ .

$y = \frac{38 \pm 3 \sqrt{19}}{19}$

$x = \frac{6 \sqrt{38}}{19}$

$y = \frac{38 \pm 3 \sqrt{19}}{19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.9

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.2.9.1

Simplify the numerator.

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Step 4.2.9.1.1

Raise $- 76$ to the power of $2$ .

$y = \frac{76 \pm \sqrt{5776 - 4 \cdot 19 \cdot 67}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.9.1.2

Multiply $- 4 \cdot 19 \cdot 67$ .

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Step 4.2.9.1.2.1

Multiply $- 4$ by $19$ .

$y = \frac{76 \pm \sqrt{5776 - 76 \cdot 67}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.9.1.2.2

Multiply $- 76$ by $67$ .

$y = \frac{76 \pm \sqrt{5776 - 5092}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm \sqrt{5776 - 5092}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.9.1.3

Subtract $5092$ from $5776$ .

$y = \frac{76 \pm \sqrt{684}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.9.1.4

Rewrite $684$ as $6^{2} \cdot 19$ .

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Step 4.2.9.1.4.1

Factor $36$ out of $684$ .

$y = \frac{76 \pm \sqrt{36 (19)}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.9.1.4.2

Rewrite $36$ as $6^{2}$ .

$y = \frac{76 \pm \sqrt{6^{2} \cdot 19}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm \sqrt{6^{2} \cdot 19}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.9.1.5

Pull terms out from under the radical.

$y = \frac{76 \pm 6 \sqrt{19}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm 6 \sqrt{19}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.9.2

Multiply $2$ by $19$ .

$y = \frac{76 \pm 6 \sqrt{19}}{38}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.9.3

Simplify $\frac{76 \pm 6 \sqrt{19}}{38}$ .

$y = \frac{38 \pm 3 \sqrt{19}}{19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.9.4

Change the $\pm$ to $+$ .

$y = \frac{38 + 3 \sqrt{19}}{19}$

$x = \frac{6 \sqrt{38}}{19}$

$y = \frac{38 + 3 \sqrt{19}}{19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.10

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 4.2.10.1

Simplify the numerator.

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Step 4.2.10.1.1

Raise $- 76$ to the power of $2$ .

$y = \frac{76 \pm \sqrt{5776 - 4 \cdot 19 \cdot 67}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.10.1.2

Multiply $- 4 \cdot 19 \cdot 67$ .

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Step 4.2.10.1.2.1

Multiply $- 4$ by $19$ .

$y = \frac{76 \pm \sqrt{5776 - 76 \cdot 67}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.10.1.2.2

Multiply $- 76$ by $67$ .

$y = \frac{76 \pm \sqrt{5776 - 5092}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm \sqrt{5776 - 5092}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.10.1.3

Subtract $5092$ from $5776$ .

$y = \frac{76 \pm \sqrt{684}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.10.1.4

Rewrite $684$ as $6^{2} \cdot 19$ .

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Step 4.2.10.1.4.1

Factor $36$ out of $684$ .

$y = \frac{76 \pm \sqrt{36 (19)}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.10.1.4.2

Rewrite $36$ as $6^{2}$ .

$y = \frac{76 \pm \sqrt{6^{2} \cdot 19}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm \sqrt{6^{2} \cdot 19}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.10.1.5

Pull terms out from under the radical.

$y = \frac{76 \pm 6 \sqrt{19}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm 6 \sqrt{19}}{2 \cdot 19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.10.2

Multiply $2$ by $19$ .

$y = \frac{76 \pm 6 \sqrt{19}}{38}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.10.3

Simplify $\frac{76 \pm 6 \sqrt{19}}{38}$ .

$y = \frac{38 \pm 3 \sqrt{19}}{19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.10.4

Change the $\pm$ to $-$ .

$y = \frac{38 - 3 \sqrt{19}}{19}$

$x = \frac{6 \sqrt{38}}{19}$

$y = \frac{38 - 3 \sqrt{19}}{19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.2.11

The final answer is the combination of both solutions.

$y = \frac{38 + 3 \sqrt{19}}{19}, \frac{38 - 3 \sqrt{19}}{19}$

$x = \frac{6 \sqrt{38}}{19}$

$y = \frac{38 + 3 \sqrt{19}}{19}, \frac{38 - 3 \sqrt{19}}{19}$

$x = \frac{6 \sqrt{38}}{19}$

Step 4.3

Solve the system of equations.

$y = \frac{38 + 3 \sqrt{19}}{19}, x = \frac{6 \sqrt{38}}{19}$

Step 4.4

Solve the system of equations.

$y = \frac{38 - 3 \sqrt{19}}{19}, x = \frac{6 \sqrt{38}}{19}$

Step 5

Solve the system $x = - \frac{6 \sqrt{38}}{19}, 4 {(y - 2)}^{2} = 36 - 9 x^{2}$ .

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Step 5.1

Replace all occurrences of $x$ with $- \frac{6 \sqrt{38}}{19}$ in each equation.

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Step 5.1.1

Replace all occurrences of $x$ in $4 {(y - 2)}^{2} = 36 - 9 x^{2}$ with $- \frac{6 \sqrt{38}}{19}$ .

$4 {(y - 2)}^{2} = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2

Simplify $4 {(y - 2)}^{2} = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$ .

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Step 5.1.2.1

Simplify the left side.

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Step 5.1.2.1.1

Simplify $4 {(y - 2)}^{2}$ .

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Step 5.1.2.1.1.1

Rewrite ${(y - 2)}^{2}$ as $(y - 2) (y - 2)$ .

$4 ((y - 2) (y - 2)) = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.1.1.2

Expand $(y - 2) (y - 2)$ using the FOIL Method.

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Step 5.1.2.1.1.2.1

Apply the distributive property.

$4 (y (y - 2) - 2 (y - 2)) = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.1.1.2.2

Apply the distributive property.

$4 (y \cdot y + y \cdot - 2 - 2 (y - 2)) = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.1.1.2.3

Apply the distributive property.

$4 (y \cdot y + y \cdot - 2 - 2 y - 2 \cdot - 2) = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

$4 (y \cdot y + y \cdot - 2 - 2 y - 2 \cdot - 2) = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.1.1.3

Simplify and combine like terms.

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Step 5.1.2.1.1.3.1

Simplify each term.

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Step 5.1.2.1.1.3.1.1

Multiply $y$ by $y$ .

$4 (y^{2} + y \cdot - 2 - 2 y - 2 \cdot - 2) = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.1.1.3.1.2

Move $- 2$ to the left of $y$ .

$4 (y^{2} - 2 \cdot y - 2 y - 2 \cdot - 2) = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.1.1.3.1.3

Multiply $- 2$ by $- 2$ .

$4 (y^{2} - 2 y - 2 y + 4) = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

$4 (y^{2} - 2 y - 2 y + 4) = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.1.1.3.2

Subtract $2 y$ from $- 2 y$ .

$4 (y^{2} - 4 y + 4) = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

$4 (y^{2} - 4 y + 4) = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.1.1.4

Apply the distributive property.

$4 y^{2} + 4 (- 4 y) + 4 \cdot 4 = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.1.1.5

Simplify.

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Step 5.1.2.1.1.5.1

Multiply $- 4$ by $4$ .

$4 y^{2} - 16 y + 4 \cdot 4 = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.1.1.5.2

Multiply $4$ by $4$ .

$4 y^{2} - 16 y + 16 = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2

Simplify the right side.

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Step 5.1.2.2.1

Simplify $36 - 9 {(- \frac{6 \sqrt{38}}{19})}^{2}$ .

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Step 5.1.2.2.1.1

Simplify each term.

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Step 5.1.2.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1.2.2.1.1.1.1

Apply the product rule to $- \frac{6 \sqrt{38}}{19}$ .

$4 y^{2} - 16 y + 16 = 36 - 9 ({(- 1)}^{2} {(\frac{6 \sqrt{38}}{19})}^{2})$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.1.2

Apply the product rule to $\frac{6 \sqrt{38}}{19}$ .

$4 y^{2} - 16 y + 16 = 36 - 9 ({(- 1)}^{2} (\frac{{(6 \sqrt{38})}^{2}}{19^{2}}))$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.1.3

Apply the product rule to $6 \sqrt{38}$ .

$4 y^{2} - 16 y + 16 = 36 - 9 ({(- 1)}^{2} (\frac{6^{2} {\sqrt{38}}^{2}}{19^{2}}))$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 ({(- 1)}^{2} (\frac{6^{2} {\sqrt{38}}^{2}}{19^{2}}))$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.2

Raise $- 1$ to the power of $2$ .

$4 y^{2} - 16 y + 16 = 36 - 9 (1 (\frac{6^{2} {\sqrt{38}}^{2}}{19^{2}}))$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.3

Multiply $\frac{6^{2} {\sqrt{38}}^{2}}{19^{2}}$ by $1$ .

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{6^{2} {\sqrt{38}}^{2}}{19^{2}}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.4

Simplify the numerator.

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Step 5.1.2.2.1.1.4.1

Raise $6$ to the power of $2$ .

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 {\sqrt{38}}^{2}}{19^{2}}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.4.2

Rewrite ${\sqrt{38}}^{2}$ as $38$ .

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Step 5.1.2.2.1.1.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{38}$ as $38^{\frac{1}{2}}$ .

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 {(38^{\frac{1}{2}})}^{2}}{19^{2}}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.4.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38^{\frac{1}{2} \cdot 2}}{19^{2}}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.4.2.3

Combine $\frac{1}{2}$ and $2$ .

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38^{\frac{2}{2}}}{19^{2}}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.4.2.4

Cancel the common factor of $2$ .

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Step 5.1.2.2.1.1.4.2.4.1

Cancel the common factor.

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38^{\frac{2}{2}}}{19^{2}}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.4.2.4.2

Rewrite the expression.

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38}{19^{2}}$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38}{19^{2}}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.4.2.5

Evaluate the exponent.

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38}{19^{2}}$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38}{19^{2}}$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38}{19^{2}}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.5

Raise $19$ to the power of $2$ .

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{36 \cdot 38}{361}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.6

Multiply $36$ by $38$ .

$4 y^{2} - 16 y + 16 = 36 - 9 (\frac{1368}{361})$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.7

Cancel the common factor of $1368$ and $361$ .

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Step 5.1.2.2.1.1.7.1

Factor $19$ out of $1368$ .

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{19 (72)}{361}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.7.2

Cancel the common factors.

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Step 5.1.2.2.1.1.7.2.1

Factor $19$ out of $361$ .

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{19 \cdot 72}{19 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.7.2.2

Cancel the common factor.

$4 y^{2} - 16 y + 16 = 36 - 9 \frac{19 \cdot 72}{19 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.7.2.3

Rewrite the expression.

$4 y^{2} - 16 y + 16 = 36 - 9 (\frac{72}{19})$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 (\frac{72}{19})$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - 9 (\frac{72}{19})$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.8

Multiply $- 9 (\frac{72}{19})$ .

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Step 5.1.2.2.1.1.8.1

Combine $- 9$ and $\frac{72}{19}$ .

$4 y^{2} - 16 y + 16 = 36 + \frac{- 9 \cdot 72}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.8.2

Multiply $- 9$ by $72$ .

$4 y^{2} - 16 y + 16 = 36 + \frac{- 648}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 + \frac{- 648}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.1.9

Move the negative in front of the fraction.

$4 y^{2} - 16 y + 16 = 36 - \frac{648}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = 36 - \frac{648}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.2

To write $36$ as a fraction with a common denominator, multiply by $\frac{19}{19}$ .

$4 y^{2} - 16 y + 16 = 36 \cdot \frac{19}{19} - \frac{648}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.3

Combine $36$ and $\frac{19}{19}$ .

$4 y^{2} - 16 y + 16 = \frac{36 \cdot 19}{19} - \frac{648}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.4

Combine the numerators over the common denominator.

$4 y^{2} - 16 y + 16 = \frac{36 \cdot 19 - 648}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.5

Simplify the numerator.

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Step 5.1.2.2.1.5.1

Multiply $36$ by $19$ .

$4 y^{2} - 16 y + 16 = \frac{684 - 648}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.1.2.2.1.5.2

Subtract $648$ from $684$ .

$4 y^{2} - 16 y + 16 = \frac{36}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = \frac{36}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = \frac{36}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = \frac{36}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = \frac{36}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + 16 = \frac{36}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2

Solve for $y$ in $4 y^{2} - 16 y + 16 = \frac{36}{19}$ .

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Step 5.2.1

Subtract $\frac{36}{19}$ from both sides of the equation.

$4 y^{2} - 16 y + 16 - \frac{36}{19} = 0$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.2

Simplify $4 y^{2} - 16 y + 16 - \frac{36}{19}$ .

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Step 5.2.2.1

To write $16$ as a fraction with a common denominator, multiply by $\frac{19}{19}$ .

$4 y^{2} - 16 y + 16 \cdot \frac{19}{19} - \frac{36}{19} = 0$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.2.2

Combine $16$ and $\frac{19}{19}$ .

$4 y^{2} - 16 y + \frac{16 \cdot 19}{19} - \frac{36}{19} = 0$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.2.3

Combine the numerators over the common denominator.

$4 y^{2} - 16 y + \frac{16 \cdot 19 - 36}{19} = 0$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.2.4

Simplify the numerator.

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Step 5.2.2.4.1

Multiply $16$ by $19$ .

$4 y^{2} - 16 y + \frac{304 - 36}{19} = 0$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.2.4.2

Subtract $36$ from $304$ .

$4 y^{2} - 16 y + \frac{268}{19} = 0$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + \frac{268}{19} = 0$

$x = - \frac{6 \sqrt{38}}{19}$

$4 y^{2} - 16 y + \frac{268}{19} = 0$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.3

Factor $4$ out of $4 y^{2} - 16 y + \frac{268}{19}$ .

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Step 5.2.3.1

Factor $4$ out of $4 y^{2}$ .

$4 (y^{2}) - 16 y + \frac{268}{19} = 0$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.3.2

Factor $4$ out of $- 16 y$ .

$4 (y^{2}) + 4 (- 4 y) + \frac{268}{19} = 0$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.3.3

Factor $4$ out of $\frac{268}{19}$ .

$4 y^{2} + 4 (- 4 y) + 4 (\frac{67}{19}) = 0$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.3.4

Factor $4$ out of $4 y^{2} + 4 (- 4 y)$ .

$4 (y^{2} - 4 y) + 4 (\frac{67}{19}) = 0$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.3.5

Factor $4$ out of $4 (y^{2} - 4 y) + 4 (\frac{67}{19})$ .

$4 (y^{2} - 4 y + \frac{67}{19}) = 0$

$x = - \frac{6 \sqrt{38}}{19}$

$4 (y^{2} - 4 y + \frac{67}{19}) = 0$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.4

Divide each term in $4 (y^{2} - 4 y + \frac{67}{19}) = 0$ by $4$ and simplify.

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Step 5.2.4.1

Divide each term in $4 (y^{2} - 4 y + \frac{67}{19}) = 0$ by $4$ .

$\frac{4 (y^{2} - 4 y + \frac{67}{19})}{4} = \frac{0}{4}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.4.2

Simplify the left side.

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Step 5.2.4.2.1

Cancel the common factor of $4$ .

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Step 5.2.4.2.1.1

Cancel the common factor.

$\frac{4 (y^{2} - 4 y + \frac{67}{19})}{4} = \frac{0}{4}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.4.2.1.2

Divide $y^{2} - 4 y + \frac{67}{19}$ by $1$ .

$y^{2} - 4 y + \frac{67}{19} = \frac{0}{4}$

$x = - \frac{6 \sqrt{38}}{19}$

$y^{2} - 4 y + \frac{67}{19} = \frac{0}{4}$

$x = - \frac{6 \sqrt{38}}{19}$

$y^{2} - 4 y + \frac{67}{19} = \frac{0}{4}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.4.3

Simplify the right side.

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Step 5.2.4.3.1

Divide $0$ by $4$ .

$y^{2} - 4 y + \frac{67}{19} = 0$

$x = - \frac{6 \sqrt{38}}{19}$

$y^{2} - 4 y + \frac{67}{19} = 0$

$x = - \frac{6 \sqrt{38}}{19}$

$y^{2} - 4 y + \frac{67}{19} = 0$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.5

Multiply through by the least common denominator $19$ , then simplify.

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Step 5.2.5.1

Apply the distributive property.

$19 y^{2} + 19 (- 4 y) + 19 (\frac{67}{19}) = 0$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.5.2

Simplify.

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Step 5.2.5.2.1

Multiply $- 4$ by $19$ .

$19 y^{2} - 76 y + 19 (\frac{67}{19}) = 0$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.5.2.2

Cancel the common factor of $19$ .

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Step 5.2.5.2.2.1

Cancel the common factor.

$19 y^{2} - 76 y + 19 (\frac{67}{19}) = 0$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.5.2.2.2

Rewrite the expression.

$19 y^{2} - 76 y + 67 = 0$

$x = - \frac{6 \sqrt{38}}{19}$

$19 y^{2} - 76 y + 67 = 0$

$x = - \frac{6 \sqrt{38}}{19}$

$19 y^{2} - 76 y + 67 = 0$

$x = - \frac{6 \sqrt{38}}{19}$

$19 y^{2} - 76 y + 67 = 0$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.6

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.7

Substitute the values $a = 19$ , $b = - 76$ , and $c = 67$ into the quadratic formula and solve for $y$ .

$\frac{76 \pm \sqrt{{(- 76)}^{2} - 4 \cdot (19 \cdot 67)}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.8

Simplify.

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Step 5.2.8.1

Simplify the numerator.

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Step 5.2.8.1.1

Raise $- 76$ to the power of $2$ .

$y = \frac{76 \pm \sqrt{5776 - 4 \cdot 19 \cdot 67}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.8.1.2

Multiply $- 4 \cdot 19 \cdot 67$ .

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Step 5.2.8.1.2.1

Multiply $- 4$ by $19$ .

$y = \frac{76 \pm \sqrt{5776 - 76 \cdot 67}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.8.1.2.2

Multiply $- 76$ by $67$ .

$y = \frac{76 \pm \sqrt{5776 - 5092}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm \sqrt{5776 - 5092}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.8.1.3

Subtract $5092$ from $5776$ .

$y = \frac{76 \pm \sqrt{684}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.8.1.4

Rewrite $684$ as $6^{2} \cdot 19$ .

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Step 5.2.8.1.4.1

Factor $36$ out of $684$ .

$y = \frac{76 \pm \sqrt{36 (19)}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.8.1.4.2

Rewrite $36$ as $6^{2}$ .

$y = \frac{76 \pm \sqrt{6^{2} \cdot 19}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm \sqrt{6^{2} \cdot 19}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.8.1.5

Pull terms out from under the radical.

$y = \frac{76 \pm 6 \sqrt{19}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm 6 \sqrt{19}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.8.2

Multiply $2$ by $19$ .

$y = \frac{76 \pm 6 \sqrt{19}}{38}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.8.3

Simplify $\frac{76 \pm 6 \sqrt{19}}{38}$ .

$y = \frac{38 \pm 3 \sqrt{19}}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

$y = \frac{38 \pm 3 \sqrt{19}}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.9

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 5.2.9.1

Simplify the numerator.

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Step 5.2.9.1.1

Raise $- 76$ to the power of $2$ .

$y = \frac{76 \pm \sqrt{5776 - 4 \cdot 19 \cdot 67}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.9.1.2

Multiply $- 4 \cdot 19 \cdot 67$ .

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Step 5.2.9.1.2.1

Multiply $- 4$ by $19$ .

$y = \frac{76 \pm \sqrt{5776 - 76 \cdot 67}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.9.1.2.2

Multiply $- 76$ by $67$ .

$y = \frac{76 \pm \sqrt{5776 - 5092}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm \sqrt{5776 - 5092}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.9.1.3

Subtract $5092$ from $5776$ .

$y = \frac{76 \pm \sqrt{684}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.9.1.4

Rewrite $684$ as $6^{2} \cdot 19$ .

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Step 5.2.9.1.4.1

Factor $36$ out of $684$ .

$y = \frac{76 \pm \sqrt{36 (19)}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.9.1.4.2

Rewrite $36$ as $6^{2}$ .

$y = \frac{76 \pm \sqrt{6^{2} \cdot 19}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm \sqrt{6^{2} \cdot 19}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.9.1.5

Pull terms out from under the radical.

$y = \frac{76 \pm 6 \sqrt{19}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm 6 \sqrt{19}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.9.2

Multiply $2$ by $19$ .

$y = \frac{76 \pm 6 \sqrt{19}}{38}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.9.3

Simplify $\frac{76 \pm 6 \sqrt{19}}{38}$ .

$y = \frac{38 \pm 3 \sqrt{19}}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.9.4

Change the $\pm$ to $+$ .

$y = \frac{38 + 3 \sqrt{19}}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

$y = \frac{38 + 3 \sqrt{19}}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.10

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 5.2.10.1

Simplify the numerator.

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Step 5.2.10.1.1

Raise $- 76$ to the power of $2$ .

$y = \frac{76 \pm \sqrt{5776 - 4 \cdot 19 \cdot 67}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.10.1.2

Multiply $- 4 \cdot 19 \cdot 67$ .

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Step 5.2.10.1.2.1

Multiply $- 4$ by $19$ .

$y = \frac{76 \pm \sqrt{5776 - 76 \cdot 67}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.10.1.2.2

Multiply $- 76$ by $67$ .

$y = \frac{76 \pm \sqrt{5776 - 5092}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm \sqrt{5776 - 5092}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.10.1.3

Subtract $5092$ from $5776$ .

$y = \frac{76 \pm \sqrt{684}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.10.1.4

Rewrite $684$ as $6^{2} \cdot 19$ .

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Step 5.2.10.1.4.1

Factor $36$ out of $684$ .

$y = \frac{76 \pm \sqrt{36 (19)}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.10.1.4.2

Rewrite $36$ as $6^{2}$ .

$y = \frac{76 \pm \sqrt{6^{2} \cdot 19}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm \sqrt{6^{2} \cdot 19}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.10.1.5

Pull terms out from under the radical.

$y = \frac{76 \pm 6 \sqrt{19}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

$y = \frac{76 \pm 6 \sqrt{19}}{2 \cdot 19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.10.2

Multiply $2$ by $19$ .

$y = \frac{76 \pm 6 \sqrt{19}}{38}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.10.3

Simplify $\frac{76 \pm 6 \sqrt{19}}{38}$ .

$y = \frac{38 \pm 3 \sqrt{19}}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.10.4

Change the $\pm$ to $-$ .

$y = \frac{38 - 3 \sqrt{19}}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

$y = \frac{38 - 3 \sqrt{19}}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.2.11

The final answer is the combination of both solutions.

$y = \frac{38 + 3 \sqrt{19}}{19}, \frac{38 - 3 \sqrt{19}}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

$y = \frac{38 + 3 \sqrt{19}}{19}, \frac{38 - 3 \sqrt{19}}{19}$

$x = - \frac{6 \sqrt{38}}{19}$

Step 5.3

Solve the system of equations.

$y = \frac{38 + 3 \sqrt{19}}{19}, x = - \frac{6 \sqrt{38}}{19}$

Step 5.4

Solve the system of equations.

$y = \frac{38 - 3 \sqrt{19}}{19}, x = - \frac{6 \sqrt{38}}{19}$

Step 6

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(\frac{6 \sqrt{38}}{19}, \frac{38 + 3 \sqrt{19}}{19})$

$(\frac{6 \sqrt{38}}{19}, \frac{38 - 3 \sqrt{19}}{19})$

$(- \frac{6 \sqrt{38}}{19}, \frac{38 + 3 \sqrt{19}}{19})$

$(- \frac{6 \sqrt{38}}{19}, \frac{38 - 3 \sqrt{19}}{19})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(\frac{6 \sqrt{38}}{19}, \frac{38 + 3 \sqrt{19}}{19}), (\frac{6 \sqrt{38}}{19}, \frac{38 - 3 \sqrt{19}}{19}), (- \frac{6 \sqrt{38}}{19}, \frac{38 + 3 \sqrt{19}}{19}), (- \frac{6 \sqrt{38}}{19}, \frac{38 - 3 \sqrt{19}}{19})$

Equation Form:

$x = \frac{6 \sqrt{38}}{19}, y = \frac{38 + 3 \sqrt{19}}{19}$

$x = \frac{6 \sqrt{38}}{19}, y = \frac{38 - 3 \sqrt{19}}{19}$

$x = - \frac{6 \sqrt{38}}{19}, y = \frac{38 + 3 \sqrt{19}}{19}$

$x = - \frac{6 \sqrt{38}}{19}, y = \frac{38 - 3 \sqrt{19}}{19}$

Step 8