Algebra Examples

$y = x + 3$ $y = x^{2} + 2 x - 4$

Step 1

Eliminate the equal sides of each equation and combine.

$x + 3 = x^{2} + 2 x - 4$

Step 2

Solve $x + 3 = x^{2} + 2 x - 4$ for $x$ .

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Step 2.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 2 x - 4 = x + 3$

Step 2.2

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

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Step 2.2.1

Subtract $x$ from both sides of the equation.

$x^{2} + 2 x - 4 - x = 3$

Step 2.2.2

Subtract $x$ from $2 x$ .

$x^{2} + x - 4 = 3$

Step 2.3

Move all terms to the left side of the equation and simplify.

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Step 2.3.1

Subtract $3$ from both sides of the equation.

$x^{2} + x - 4 - 3 = 0$

Step 2.3.2

Subtract $3$ from $- 4$ .

$x^{2} + x - 7 = 0$

Step 2.4

Use the quadratic formula to find the solutions.

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

Step 2.5

Substitute the values $a = 1$ , $b = 1$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot - 7)}}{2 \cdot 1} + y = x^{2} + 2 x - 4$

Step 2.6

Simplify.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 7}}{2 \cdot 1}$

Step 2.6.1.2

Multiply $- 4 \cdot 1 \cdot - 7$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 7}}{2 \cdot 1}$

Step 2.6.1.2.2

Multiply $- 4$ by $- 7$ .

$x = \frac{- 1 \pm \sqrt{1 + 28}}{2 \cdot 1}$

Step 2.6.1.3

Add $1$ and $28$ .

$x = \frac{- 1 \pm \sqrt{29}}{2 \cdot 1}$

Step 2.6.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{29}}{2}$

Step 2.7

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

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 7}}{2 \cdot 1}$

Step 2.7.1.2

Multiply $- 4 \cdot 1 \cdot - 7$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 7}}{2 \cdot 1}$

Step 2.7.1.2.2

Multiply $- 4$ by $- 7$ .

$x = \frac{- 1 \pm \sqrt{1 + 28}}{2 \cdot 1}$

Step 2.7.1.3

Add $1$ and $28$ .

$x = \frac{- 1 \pm \sqrt{29}}{2 \cdot 1}$

Step 2.7.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{29}}{2}$

Step 2.7.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{29}}{2}$

Step 2.7.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + \sqrt{29}}{2}$

Step 2.7.5

Factor $- 1$ out of $\sqrt{29}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{29})}{2}$

Step 2.7.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{29})$ .

$x = \frac{- 1 (1 - \sqrt{29})}{2}$

Step 2.7.7

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{29}}{2}$

Step 2.8

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

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 7}}{2 \cdot 1}$

Step 2.8.1.2

Multiply $- 4 \cdot 1 \cdot - 7$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 7}}{2 \cdot 1}$

Step 2.8.1.2.2

Multiply $- 4$ by $- 7$ .

$x = \frac{- 1 \pm \sqrt{1 + 28}}{2 \cdot 1}$

Step 2.8.1.3

Add $1$ and $28$ .

$x = \frac{- 1 \pm \sqrt{29}}{2 \cdot 1}$

Step 2.8.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{29}}{2}$

Step 2.8.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{29}}{2}$

Step 2.8.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 - \sqrt{29}}{2}$

Step 2.8.5

Factor $- 1$ out of $- \sqrt{29}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{29})}{2}$

Step 2.8.6

Factor $- 1$ out of $- 1 (1) - (\sqrt{29})$ .

$x = \frac{- 1 (1 + \sqrt{29})}{2}$

Step 2.8.7

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{29}}{2}$

Step 2.9

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{29}}{2}, - \frac{1 + \sqrt{29}}{2}$

Step 3

Evaluate $y$ when $x = - \frac{1 - \sqrt{29}}{2}$ .

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Step 3.1

Substitute $- \frac{1 - \sqrt{29}}{2}$ for $x$ .

$y = {(- \frac{1 - \sqrt{29}}{2})}^{2} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2

Simplify ${(- \frac{1 - \sqrt{29}}{2})}^{2} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$ .

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

Apply the product rule to $- \frac{1 - \sqrt{29}}{2}$ .

$y = {(- 1)}^{2} {(\frac{1 - \sqrt{29}}{2})}^{2} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.1.2

Apply the product rule to $\frac{1 - \sqrt{29}}{2}$ .

$y = {(- 1)}^{2} \frac{{(1 - \sqrt{29})}^{2}}{2^{2}} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.2

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

$y = 1 \frac{{(1 - \sqrt{29})}^{2}}{2^{2}} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.3

Multiply $\frac{{(1 - \sqrt{29})}^{2}}{2^{2}}$ by $1$ .

$y = \frac{{(1 - \sqrt{29})}^{2}}{2^{2}} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.4

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

$y = \frac{{(1 - \sqrt{29})}^{2}}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.5

Rewrite ${(1 - \sqrt{29})}^{2}$ as $(1 - \sqrt{29}) (1 - \sqrt{29})$ .

$y = \frac{(1 - \sqrt{29}) (1 - \sqrt{29})}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.6

Expand $(1 - \sqrt{29}) (1 - \sqrt{29})$ using the FOIL Method.

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Step 3.2.1.6.1

Apply the distributive property.

$y = \frac{1 (1 - \sqrt{29}) - \sqrt{29} (1 - \sqrt{29})}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.6.2

Apply the distributive property.

$y = \frac{1 \cdot 1 + 1 (- \sqrt{29}) - \sqrt{29} (1 - \sqrt{29})}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.6.3

Apply the distributive property.

$y = \frac{1 \cdot 1 + 1 (- \sqrt{29}) - \sqrt{29} \cdot 1 - \sqrt{29} (- \sqrt{29})}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.7

Simplify and combine like terms.

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Step 3.2.1.7.1

Simplify each term.

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Step 3.2.1.7.1.1

Multiply $1$ by $1$ .

$y = \frac{1 + 1 (- \sqrt{29}) - \sqrt{29} \cdot 1 - \sqrt{29} (- \sqrt{29})}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.7.1.2

Multiply $- \sqrt{29}$ by $1$ .

$y = \frac{1 - \sqrt{29} - \sqrt{29} \cdot 1 - \sqrt{29} (- \sqrt{29})}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.7.1.3

Multiply $- 1$ by $1$ .

$y = \frac{1 - \sqrt{29} - \sqrt{29} - \sqrt{29} (- \sqrt{29})}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.7.1.4

Multiply $- \sqrt{29} (- \sqrt{29})$ .

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Step 3.2.1.7.1.4.1

Multiply $- 1$ by $- 1$ .

$y = \frac{1 - \sqrt{29} - \sqrt{29} + 1 \sqrt{29} \sqrt{29}}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.7.1.4.2

Multiply $\sqrt{29}$ by $1$ .

$y = \frac{1 - \sqrt{29} - \sqrt{29} + \sqrt{29} \sqrt{29}}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.7.1.4.3

Raise $\sqrt{29}$ to the power of $1$ .

$y = \frac{1 - \sqrt{29} - \sqrt{29} + {\sqrt{29}}^{1} \sqrt{29}}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.7.1.4.4

Raise $\sqrt{29}$ to the power of $1$ .

$y = \frac{1 - \sqrt{29} - \sqrt{29} + {\sqrt{29}}^{1} {\sqrt{29}}^{1}}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.7.1.4.5

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

$y = \frac{1 - \sqrt{29} - \sqrt{29} + {\sqrt{29}}^{1 + 1}}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.7.1.4.6

Add $1$ and $1$ .

$y = \frac{1 - \sqrt{29} - \sqrt{29} + {\sqrt{29}}^{2}}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.7.1.5

Rewrite ${\sqrt{29}}^{2}$ as $29$ .

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Step 3.2.1.7.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{29}$ as $29^{\frac{1}{2}}$ .

$y = \frac{1 - \sqrt{29} - \sqrt{29} + {(29^{\frac{1}{2}})}^{2}}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.7.1.5.2

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

$y = \frac{1 - \sqrt{29} - \sqrt{29} + 29^{\frac{1}{2} \cdot 2}}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.7.1.5.3

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

$y = \frac{1 - \sqrt{29} - \sqrt{29} + 29^{\frac{2}{2}}}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.7.1.5.4

Cancel the common factor of $2$ .

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Step 3.2.1.7.1.5.4.1

Cancel the common factor.

$y = \frac{1 - \sqrt{29} - \sqrt{29} + 29^{\frac{2}{2}}}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.7.1.5.4.2

Rewrite the expression.

$y = \frac{1 - \sqrt{29} - \sqrt{29} + 29^{1}}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.7.1.5.5

Evaluate the exponent.

$y = \frac{1 - \sqrt{29} - \sqrt{29} + 29}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.7.2

Add $1$ and $29$ .

$y = \frac{30 - \sqrt{29} - \sqrt{29}}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.7.3

Subtract $\sqrt{29}$ from $- \sqrt{29}$ .

$y = \frac{30 - 2 \sqrt{29}}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.8

Cancel the common factor of $30 - 2 \sqrt{29}$ and $4$ .

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Step 3.2.1.8.1

Factor $2$ out of $30$ .

$y = \frac{2 (15) - 2 \sqrt{29}}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.8.2

Factor $2$ out of $- 2 \sqrt{29}$ .

$y = \frac{2 (15) + 2 (- \sqrt{29})}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.8.3

Factor $2$ out of $2 (15) + 2 (- \sqrt{29})$ .

$y = \frac{2 (15 - \sqrt{29})}{4} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.8.4

Cancel the common factors.

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Step 3.2.1.8.4.1

Factor $2$ out of $4$ .

$y = \frac{2 (15 - \sqrt{29})}{2 \cdot 2} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.8.4.2

Cancel the common factor.

$y = \frac{2 (15 - \sqrt{29})}{2 \cdot 2} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.8.4.3

Rewrite the expression.

$y = \frac{15 - \sqrt{29}}{2} + 2 (- \frac{1 - \sqrt{29}}{2}) - 4$

Step 3.2.1.9

Cancel the common factor of $2$ .

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Step 3.2.1.9.1

Move the leading negative in $- \frac{1 - \sqrt{29}}{2}$ into the numerator.

$y = \frac{15 - \sqrt{29}}{2} + 2 \frac{- (1 - \sqrt{29})}{2} - 4$

Step 3.2.1.9.2

Cancel the common factor.

$y = \frac{15 - \sqrt{29}}{2} + 2 \frac{- (1 - \sqrt{29})}{2} - 4$

Step 3.2.1.9.3

Rewrite the expression.

$y = \frac{15 - \sqrt{29}}{2} - (1 - \sqrt{29}) - 4$

Step 3.2.1.10

Apply the distributive property.

$y = \frac{15 - \sqrt{29}}{2} - 1 \cdot 1 - - \sqrt{29} - 4$

Step 3.2.1.11

Multiply $- 1$ by $1$ .

$y = \frac{15 - \sqrt{29}}{2} - 1 - - \sqrt{29} - 4$

Step 3.2.1.12

Multiply $- - \sqrt{29}$ .

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Step 3.2.1.12.1

Multiply $- 1$ by $- 1$ .

$y = \frac{15 - \sqrt{29}}{2} - 1 + 1 \sqrt{29} - 4$

Step 3.2.1.12.2

Multiply $\sqrt{29}$ by $1$ .

$y = \frac{15 - \sqrt{29}}{2} - 1 + \sqrt{29} - 4$

Step 3.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{15 - \sqrt{29}}{2} - 1 \cdot \frac{2}{2} + \sqrt{29} - 4$

Step 3.2.3

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

$y = \frac{15 - \sqrt{29}}{2} + \frac{- 1 \cdot 2}{2} + \sqrt{29} - 4$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Combine the numerators over the common denominator.

$y = \frac{15 - \sqrt{29} - 1 \cdot 2}{2} + \sqrt{29} - 4$

Step 3.2.4.2

Multiply $- 1$ by $2$ .

$y = \frac{15 - \sqrt{29} - 2}{2} + \sqrt{29} - 4$

Step 3.2.4.3

Subtract $2$ from $15$ .

$y = \frac{13 - \sqrt{29}}{2} + \sqrt{29} - 4$

Step 3.2.5

To write $\sqrt{29}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{13 - \sqrt{29}}{2} + \sqrt{29} \cdot \frac{2}{2} - 4$

Step 3.2.6

Combine $\sqrt{29}$ and $\frac{2}{2}$ .

$y = \frac{13 - \sqrt{29}}{2} + \frac{\sqrt{29} \cdot 2}{2} - 4$

Step 3.2.7

Simplify the expression.

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Step 3.2.7.1

Combine the numerators over the common denominator.

$y = \frac{13 - \sqrt{29} + \sqrt{29} \cdot 2}{2} - 4$

Step 3.2.7.2

Reorder the factors of $\sqrt{29} \cdot 2$ .

$y = \frac{13 - \sqrt{29} + 2 \sqrt{29}}{2} - 4$

Step 3.2.8

Add $- \sqrt{29}$ and $2 \sqrt{29}$ .

$y = \frac{13 + \sqrt{29}}{2} - 4$

Step 3.2.9

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{13 + \sqrt{29}}{2} - 4 \cdot \frac{2}{2}$

Step 3.2.10

Combine fractions.

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Step 3.2.10.1

Combine $- 4$ and $\frac{2}{2}$ .

$y = \frac{13 + \sqrt{29}}{2} + \frac{- 4 \cdot 2}{2}$

Step 3.2.10.2

Combine the numerators over the common denominator.

$y = \frac{13 + \sqrt{29} - 4 \cdot 2}{2}$

Step 3.2.11

Simplify the numerator.

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Step 3.2.11.1

Multiply $- 4$ by $2$ .

$y = \frac{13 + \sqrt{29} - 8}{2}$

Step 3.2.11.2

Subtract $8$ from $13$ .

$y = \frac{5 + \sqrt{29}}{2}$

Step 4

Evaluate $y$ when $x = - \frac{1 + \sqrt{29}}{2}$ .

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Step 4.1

Substitute $- \frac{1 + \sqrt{29}}{2}$ for $x$ .

$y = {(- \frac{1 + \sqrt{29}}{2})}^{2} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2

Simplify ${(- \frac{1 + \sqrt{29}}{2})}^{2} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

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Step 4.2.1.1.1

Apply the product rule to $- \frac{1 + \sqrt{29}}{2}$ .

$y = {(- 1)}^{2} {(\frac{1 + \sqrt{29}}{2})}^{2} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.1.2

Apply the product rule to $\frac{1 + \sqrt{29}}{2}$ .

$y = {(- 1)}^{2} \frac{{(1 + \sqrt{29})}^{2}}{2^{2}} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.2

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

$y = 1 \frac{{(1 + \sqrt{29})}^{2}}{2^{2}} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.3

Multiply $\frac{{(1 + \sqrt{29})}^{2}}{2^{2}}$ by $1$ .

$y = \frac{{(1 + \sqrt{29})}^{2}}{2^{2}} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.4

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

$y = \frac{{(1 + \sqrt{29})}^{2}}{4} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.5

Rewrite ${(1 + \sqrt{29})}^{2}$ as $(1 + \sqrt{29}) (1 + \sqrt{29})$ .

$y = \frac{(1 + \sqrt{29}) (1 + \sqrt{29})}{4} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.6

Expand $(1 + \sqrt{29}) (1 + \sqrt{29})$ using the FOIL Method.

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Step 4.2.1.6.1

Apply the distributive property.

$y = \frac{1 (1 + \sqrt{29}) + \sqrt{29} (1 + \sqrt{29})}{4} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.6.2

Apply the distributive property.

$y = \frac{1 \cdot 1 + 1 \sqrt{29} + \sqrt{29} (1 + \sqrt{29})}{4} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.6.3

Apply the distributive property.

$y = \frac{1 \cdot 1 + 1 \sqrt{29} + \sqrt{29} \cdot 1 + \sqrt{29} \sqrt{29}}{4} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.7

Simplify and combine like terms.

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Step 4.2.1.7.1

Simplify each term.

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Step 4.2.1.7.1.1

Multiply $1$ by $1$ .

$y = \frac{1 + 1 \sqrt{29} + \sqrt{29} \cdot 1 + \sqrt{29} \sqrt{29}}{4} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.7.1.2

Multiply $\sqrt{29}$ by $1$ .

$y = \frac{1 + \sqrt{29} + \sqrt{29} \cdot 1 + \sqrt{29} \sqrt{29}}{4} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.7.1.3

Multiply $\sqrt{29}$ by $1$ .

$y = \frac{1 + \sqrt{29} + \sqrt{29} + \sqrt{29} \sqrt{29}}{4} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.7.1.4

Combine using the product rule for radicals.

$y = \frac{1 + \sqrt{29} + \sqrt{29} + \sqrt{29 \cdot 29}}{4} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.7.1.5

Multiply $29$ by $29$ .

$y = \frac{1 + \sqrt{29} + \sqrt{29} + \sqrt{841}}{4} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.7.1.6

Rewrite $841$ as $29^{2}$ .

$y = \frac{1 + \sqrt{29} + \sqrt{29} + \sqrt{29^{2}}}{4} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.7.1.7

Pull terms out from under the radical, assuming positive real numbers.

$y = \frac{1 + \sqrt{29} + \sqrt{29} + 29}{4} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.7.2

Add $1$ and $29$ .

$y = \frac{30 + \sqrt{29} + \sqrt{29}}{4} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.7.3

Add $\sqrt{29}$ and $\sqrt{29}$ .

$y = \frac{30 + 2 \sqrt{29}}{4} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.8

Cancel the common factor of $30 + 2 \sqrt{29}$ and $4$ .

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Step 4.2.1.8.1

Factor $2$ out of $30$ .

$y = \frac{2 \cdot 15 + 2 \sqrt{29}}{4} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.8.2

Factor $2$ out of $2 \sqrt{29}$ .

$y = \frac{2 \cdot 15 + 2 (\sqrt{29})}{4} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.8.3

Factor $2$ out of $2 \cdot 15 + 2 (\sqrt{29})$ .

$y = \frac{2 \cdot (15 + \sqrt{29})}{4} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.8.4

Cancel the common factors.

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Step 4.2.1.8.4.1

Factor $2$ out of $4$ .

$y = \frac{2 \cdot (15 + \sqrt{29})}{2 (2)} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.8.4.2

Cancel the common factor.

$y = \frac{2 \cdot (15 + \sqrt{29})}{2 \cdot 2} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.8.4.3

Rewrite the expression.

$y = \frac{15 + \sqrt{29}}{2} + 2 (- \frac{1 + \sqrt{29}}{2}) - 4$

Step 4.2.1.9

Cancel the common factor of $2$ .

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Step 4.2.1.9.1

Move the leading negative in $- \frac{1 + \sqrt{29}}{2}$ into the numerator.

$y = \frac{15 + \sqrt{29}}{2} + 2 \frac{- (1 + \sqrt{29})}{2} - 4$

Step 4.2.1.9.2

Cancel the common factor.

$y = \frac{15 + \sqrt{29}}{2} + 2 \frac{- (1 + \sqrt{29})}{2} - 4$

Step 4.2.1.9.3

Rewrite the expression.

$y = \frac{15 + \sqrt{29}}{2} - (1 + \sqrt{29}) - 4$

Step 4.2.1.10

Apply the distributive property.

$y = \frac{15 + \sqrt{29}}{2} - 1 \cdot 1 - \sqrt{29} - 4$

Step 4.2.1.11

Multiply $- 1$ by $1$ .

$y = \frac{15 + \sqrt{29}}{2} - 1 - \sqrt{29} - 4$

Step 4.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{15 + \sqrt{29}}{2} - 1 \cdot \frac{2}{2} - \sqrt{29} - 4$

Step 4.2.3

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

$y = \frac{15 + \sqrt{29}}{2} + \frac{- 1 \cdot 2}{2} - \sqrt{29} - 4$

Step 4.2.4

Simplify the expression.

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Step 4.2.4.1

Combine the numerators over the common denominator.

$y = \frac{15 + \sqrt{29} - 1 \cdot 2}{2} - \sqrt{29} - 4$

Step 4.2.4.2

Multiply $- 1$ by $2$ .

$y = \frac{15 + \sqrt{29} - 2}{2} - \sqrt{29} - 4$

Step 4.2.4.3

Subtract $2$ from $15$ .

$y = \frac{13 + \sqrt{29}}{2} - \sqrt{29} - 4$

Step 4.2.5

To write $- \sqrt{29}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{13 + \sqrt{29}}{2} - \sqrt{29} \cdot \frac{2}{2} - 4$

Step 4.2.6

Combine fractions.

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Step 4.2.6.1

Combine $- \sqrt{29}$ and $\frac{2}{2}$ .

$y = \frac{13 + \sqrt{29}}{2} + \frac{- \sqrt{29} \cdot 2}{2} - 4$

Step 4.2.6.2

Combine the numerators over the common denominator.

$y = \frac{13 + \sqrt{29} - \sqrt{29} \cdot 2}{2} - 4$

Step 4.2.7

Simplify the numerator.

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Step 4.2.7.1

Multiply $2$ by $- 1$ .

$y = \frac{13 + \sqrt{29} - 2 \sqrt{29}}{2} - 4$

Step 4.2.7.2

Subtract $2 \sqrt{29}$ from $\sqrt{29}$ .

$y = \frac{13 - \sqrt{29}}{2} - 4$

Step 4.2.8

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{13 - \sqrt{29}}{2} - 4 \cdot \frac{2}{2}$

Step 4.2.9

Combine fractions.

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Step 4.2.9.1

Combine $- 4$ and $\frac{2}{2}$ .

$y = \frac{13 - \sqrt{29}}{2} + \frac{- 4 \cdot 2}{2}$

Step 4.2.9.2

Combine the numerators over the common denominator.

$y = \frac{13 - \sqrt{29} - 4 \cdot 2}{2}$

Step 4.2.10

Simplify the numerator.

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Step 4.2.10.1

Multiply $- 4$ by $2$ .

$y = \frac{13 - \sqrt{29} - 8}{2}$

Step 4.2.10.2

Subtract $8$ from $13$ .

$y = \frac{5 - \sqrt{29}}{2}$

Step 5

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

$(- \frac{1 - \sqrt{29}}{2}, \frac{5 + \sqrt{29}}{2})$

$(- \frac{1 + \sqrt{29}}{2}, \frac{5 - \sqrt{29}}{2})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- \frac{1 - \sqrt{29}}{2}, \frac{5 + \sqrt{29}}{2}), (- \frac{1 + \sqrt{29}}{2}, \frac{5 - \sqrt{29}}{2})$

Equation Form:

$x = - \frac{1 - \sqrt{29}}{2}, y = \frac{5 + \sqrt{29}}{2}$

$x = - \frac{1 + \sqrt{29}}{2}, y = \frac{5 - \sqrt{29}}{2}$

Step 7