Algebra Examples

$x^{2} = - \frac{32}{3} y$ $x^{2} + y^{2} = 100$

Step 1

Solve for $x$ in $x^{2} = - \frac{32}{3} y$ .

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Step 1.1

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

$x = \pm \sqrt{- \frac{32}{3} y}$

$x^{2} + y^{2} = 100$

Step 1.2

Simplify $\pm \sqrt{- \frac{32}{3} y}$ .

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Step 1.2.1

Combine $y$ and $\frac{32}{3}$ .

$x = \pm \sqrt{- \frac{y \cdot 32}{3}}$

$x^{2} + y^{2} = 100$

Step 1.2.2

Move $32$ to the left of $y$ .

$x = \pm \sqrt{- \frac{32 y}{3}}$

$x^{2} + y^{2} = 100$

$x = \pm \sqrt{- \frac{32 y}{3}}$

$x^{2} + y^{2} = 100$

Step 1.3

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

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Step 1.3.1

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

$x = \sqrt{- \frac{32 y}{3}}$

$x^{2} + y^{2} = 100$

Step 1.3.2

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

$x = - \sqrt{- \frac{32 y}{3}}$

$x^{2} + y^{2} = 100$

Step 1.3.3

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

$x = \sqrt{- \frac{32 y}{3}}$

$x = - \sqrt{- \frac{32 y}{3}}$

$x^{2} + y^{2} = 100$

$x = \sqrt{- \frac{32 y}{3}}$

$x = - \sqrt{- \frac{32 y}{3}}$

$x^{2} + y^{2} = 100$

$x = \sqrt{- \frac{32 y}{3}}$

$x = - \sqrt{- \frac{32 y}{3}}$

$x^{2} + y^{2} = 100$

Step 2

Solve the system $x = \sqrt{- \frac{32 y}{3}}, x^{2} + y^{2} = 100$ .

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Step 2.1

Replace all occurrences of $x$ with $\sqrt{- \frac{32 y}{3}}$ in each equation.

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Step 2.1.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 100$ with $\sqrt{- \frac{32 y}{3}}$ .

${(\sqrt{- \frac{32 y}{3}})}^{2} + y^{2} = 100$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Rewrite ${\sqrt{- \frac{32 y}{3}}}^{2}$ as $- \frac{32 y}{3}$ .

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Step 2.1.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- \frac{32 y}{3}}$ as ${(- \frac{32 y}{3})}^{\frac{1}{2}}$ .

${({(- \frac{32 y}{3})}^{\frac{1}{2}})}^{2} + y^{2} = 100$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.1.2.1.2

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

${(- \frac{32 y}{3})}^{\frac{1}{2} \cdot 2} + y^{2} = 100$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.1.2.1.3

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

${(- \frac{32 y}{3})}^{\frac{2}{2}} + y^{2} = 100$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.1.2.1.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.4.1

Cancel the common factor.

${(- \frac{32 y}{3})}^{\frac{2}{2}} + y^{2} = 100$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.1.2.1.4.2

Rewrite the expression.

$(- \frac{32 y}{3}) + y^{2} = 100$

$x = \sqrt{- \frac{32 y}{3}}$

$(- \frac{32 y}{3}) + y^{2} = 100$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.1.2.1.5

Simplify.

$- \frac{32 y}{3} + y^{2} = 100$

$x = \sqrt{- \frac{32 y}{3}}$

$- \frac{32 y}{3} + y^{2} = 100$

$x = \sqrt{- \frac{32 y}{3}}$

$- \frac{32 y}{3} + y^{2} = 100$

$x = \sqrt{- \frac{32 y}{3}}$

$- \frac{32 y}{3} + y^{2} = 100$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2

Solve for $y$ in $- \frac{32 y}{3} + y^{2} = 100$ .

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Step 2.2.1

Multiply each term in $- \frac{32 y}{3} + y^{2} = 100$ by $3$ to eliminate the fractions.

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Step 2.2.1.1

Multiply each term in $- \frac{32 y}{3} + y^{2} = 100$ by $3$ .

$- \frac{32 y}{3} \cdot 3 + y^{2} \cdot 3 = 100 \cdot 3$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.1.2

Simplify the left side.

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Step 2.2.1.2.1

Simplify each term.

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Step 2.2.1.2.1.1

Cancel the common factor of $3$ .

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Step 2.2.1.2.1.1.1

Move the leading negative in $- \frac{32 y}{3}$ into the numerator.

$\frac{- 32 y}{3} \cdot 3 + y^{2} \cdot 3 = 100 \cdot 3$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.1.2.1.1.2

Cancel the common factor.

$\frac{- 32 y}{3} \cdot 3 + y^{2} \cdot 3 = 100 \cdot 3$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.1.2.1.1.3

Rewrite the expression.

$- 32 y + y^{2} \cdot 3 = 100 \cdot 3$

$x = \sqrt{- \frac{32 y}{3}}$

$- 32 y + y^{2} \cdot 3 = 100 \cdot 3$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.1.2.1.2

Move $3$ to the left of $y^{2}$ .

$- 32 y + 3 y^{2} = 100 \cdot 3$

$x = \sqrt{- \frac{32 y}{3}}$

$- 32 y + 3 y^{2} = 100 \cdot 3$

$x = \sqrt{- \frac{32 y}{3}}$

$- 32 y + 3 y^{2} = 100 \cdot 3$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.1.3

Simplify the right side.

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Step 2.2.1.3.1

Multiply $100$ by $3$ .

$- 32 y + 3 y^{2} = 300$

$x = \sqrt{- \frac{32 y}{3}}$

$- 32 y + 3 y^{2} = 300$

$x = \sqrt{- \frac{32 y}{3}}$

$- 32 y + 3 y^{2} = 300$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.2

Subtract $300$ from both sides of the equation.

$- 32 y + 3 y^{2} - 300 = 0$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.3

Factor the left side of the equation.

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Step 2.2.3.1

Let $u = y$ . Substitute $u$ for all occurrences of $y$ .

$- 32 u + 3 u^{2} - 300 = 0$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.3.2

Factor by grouping.

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Step 2.2.3.2.1

Reorder terms.

$3 u^{2} - 32 u - 300 = 0$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.3.2.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 300 = - 900$ and whose sum is $b = - 32$ .

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Step 2.2.3.2.2.1

Factor $- 32$ out of $- 32 u$ .

$3 u^{2} - 32 u - 300 = 0$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.3.2.2.2

Rewrite $- 32$ as $18$ plus $- 50$

$3 u^{2} + (18 - 50) u - 300 = 0$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.3.2.2.3

Apply the distributive property.

$3 u^{2} + 18 u - 50 u - 300 = 0$

$x = \sqrt{- \frac{32 y}{3}}$

$3 u^{2} + 18 u - 50 u - 300 = 0$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.3.2.3

Factor out the greatest common factor from each group.

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Step 2.2.3.2.3.1

Group the first two terms and the last two terms.

$(3 u^{2} + 18 u) - 50 u - 300 = 0$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.3.2.3.2

Factor out the greatest common factor (GCF) from each group.

$3 u (u + 6) - 50 (u + 6) = 0$

$x = \sqrt{- \frac{32 y}{3}}$

$3 u (u + 6) - 50 (u + 6) = 0$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.3.2.4

Factor the polynomial by factoring out the greatest common factor, $u + 6$ .

$(u + 6) (3 u - 50) = 0$

$x = \sqrt{- \frac{32 y}{3}}$

$(u + 6) (3 u - 50) = 0$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.3.3

Replace all occurrences of $u$ with $y$ .

$(y + 6) (3 y - 50) = 0$

$x = \sqrt{- \frac{32 y}{3}}$

$(y + 6) (3 y - 50) = 0$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y + 6 = 0$

$3 y - 50 = 0$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.5

Set $y + 6$ equal to $0$ and solve for $y$ .

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Step 2.2.5.1

Set $y + 6$ equal to $0$ .

$y + 6 = 0$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.5.2

Subtract $6$ from both sides of the equation.

$y = - 6$

$x = \sqrt{- \frac{32 y}{3}}$

$y = - 6$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.6

Set $3 y - 50$ equal to $0$ and solve for $y$ .

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Step 2.2.6.1

Set $3 y - 50$ equal to $0$ .

$3 y - 50 = 0$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.6.2

Solve $3 y - 50 = 0$ for $y$ .

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Step 2.2.6.2.1

Add $50$ to both sides of the equation.

$3 y = 50$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.6.2.2

Divide each term in $3 y = 50$ by $3$ and simplify.

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Step 2.2.6.2.2.1

Divide each term in $3 y = 50$ by $3$ .

$\frac{3 y}{3} = \frac{50}{3}$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.6.2.2.2

Simplify the left side.

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Step 2.2.6.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.6.2.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{50}{3}$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.6.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{50}{3}$

$x = \sqrt{- \frac{32 y}{3}}$

$y = \frac{50}{3}$

$x = \sqrt{- \frac{32 y}{3}}$

$y = \frac{50}{3}$

$x = \sqrt{- \frac{32 y}{3}}$

$y = \frac{50}{3}$

$x = \sqrt{- \frac{32 y}{3}}$

$y = \frac{50}{3}$

$x = \sqrt{- \frac{32 y}{3}}$

$y = \frac{50}{3}$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.2.7

The final solution is all the values that make $(y + 6) (3 y - 50) = 0$ true.

$y = - 6, \frac{50}{3}$

$x = \sqrt{- \frac{32 y}{3}}$

$y = - 6, \frac{50}{3}$

$x = \sqrt{- \frac{32 y}{3}}$

Step 2.3

Replace all occurrences of $y$ with $- 6$ in each equation.

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Step 2.3.1

Replace all occurrences of $y$ in $x = \sqrt{- \frac{32 y}{3}}$ with $- 6$ .

$x = \sqrt{- \frac{32 (- 6)}{3}}$

$y = - 6$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{- \frac{32 (- 6)}{3}}$ .

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Step 2.3.2.1.1

Reduce the expression by cancelling the common factors.

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Step 2.3.2.1.1.1

Reduce the expression $\frac{32 (- 6)}{3}$ by cancelling the common factors.

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Step 2.3.2.1.1.1.1

Factor $3$ out of $32 (- 6)$ .

$x = \sqrt{- \frac{3 (32 \cdot (- 2))}{3}}$

$y = - 6$

Step 2.3.2.1.1.1.2

Factor $3$ out of $3$ .

$x = \sqrt{- \frac{3 (32 \cdot (- 2))}{3 (1)}}$

$y = - 6$

Step 2.3.2.1.1.1.3

Cancel the common factor.

$x = \sqrt{- \frac{3 (32 \cdot (- 2))}{3 \cdot 1}}$

$y = - 6$

Step 2.3.2.1.1.1.4

Rewrite the expression.

$x = \sqrt{- \frac{32 \cdot (- 2)}{1}}$

$y = - 6$

$x = \sqrt{- \frac{32 \cdot (- 2)}{1}}$

$y = - 6$

Step 2.3.2.1.1.2

Divide $32 \cdot (- 2)$ by $1$ .

$x = \sqrt{- (32 \cdot (- 2))}$

$y = - 6$

$x = \sqrt{- 1 \cdot 32 \cdot - 2}$

$y = - 6$

Step 2.3.2.1.2

Combine exponents.

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Step 2.3.2.1.2.1

Multiply $- 1$ by $32$ .

$x = \sqrt{- 32 \cdot - 2}$

$y = - 6$

Step 2.3.2.1.2.2

Multiply $- 32$ by $- 2$ .

$x = \sqrt{64}$

$y = - 6$

$x = \sqrt{64}$

$y = - 6$

Step 2.3.2.1.3

Rewrite $64$ as $8^{2}$ .

$x = \sqrt{8^{2}}$

$y = - 6$

Step 2.3.2.1.4

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

$x = 8$

$y = - 6$

$x = 8$

$y = - 6$

$x = 8$

$y = - 6$

$x = 8$

$y = - 6$

Step 2.4

Replace all occurrences of $y$ with $\frac{50}{3}$ in each equation.

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Step 2.4.1

Replace all occurrences of $y$ in $x = \sqrt{- \frac{32 y}{3}}$ with $\frac{50}{3}$ .

$x = \sqrt{- \frac{32 (\frac{50}{3})}{3}}$

$y = \frac{50}{3}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\sqrt{- \frac{32 (\frac{50}{3})}{3}}$ .

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Step 2.4.2.1.1

Combine $32$ and $\frac{50}{3}$ .

$x = \sqrt{- \frac{\frac{32 \cdot 50}{3}}{3}}$

$y = \frac{50}{3}$

Step 2.4.2.1.2

Multiply $32$ by $50$ .

$x = \sqrt{- \frac{\frac{1600}{3}}{3}}$

$y = \frac{50}{3}$

Step 2.4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$x = \sqrt{- (\frac{1600}{3} \cdot \frac{1}{3})}$

$y = \frac{50}{3}$

Step 2.4.2.1.4

Multiply $\frac{1600}{3} \cdot \frac{1}{3}$ .

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Step 2.4.2.1.4.1

Multiply $\frac{1600}{3}$ by $\frac{1}{3}$ .

$x = \sqrt{- \frac{1600}{3 \cdot 3}}$

$y = \frac{50}{3}$

Step 2.4.2.1.4.2

Multiply $3$ by $3$ .

$x = \sqrt{- \frac{1600}{9}}$

$y = \frac{50}{3}$

$x = \sqrt{- \frac{1600}{9}}$

$y = \frac{50}{3}$

Step 2.4.2.1.5

Rewrite $- \frac{1600}{9}$ as ${(\frac{40 i}{3})}^{2}$ .

$x = \sqrt{{(\frac{40 i}{3})}^{2}}$

$y = \frac{50}{3}$

Step 2.4.2.1.6

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

$x = \frac{40 i}{3}$

$y = \frac{50}{3}$

$x = \frac{40 i}{3}$

$y = \frac{50}{3}$

$x = \frac{40 i}{3}$

$y = \frac{50}{3}$

$x = \frac{40 i}{3}$

$y = \frac{50}{3}$

$x = \frac{40 i}{3}$

$y = \frac{50}{3}$

Step 3

Solve the system $x = - \sqrt{- \frac{32 y}{3}}, x^{2} + y^{2} = 100$ .

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Step 3.1

Replace all occurrences of $x$ with $- \sqrt{- \frac{32 y}{3}}$ in each equation.

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Step 3.1.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 100$ with $- \sqrt{- \frac{32 y}{3}}$ .

${(- \sqrt{- \frac{32 y}{3}})}^{2} + y^{2} = 100$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Apply the product rule to $- \sqrt{- \frac{32 y}{3}}$ .

${(- 1)}^{2} {\sqrt{- \frac{32 y}{3}}}^{2} + y^{2} = 100$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.1.2.1.2

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

$1 {\sqrt{- \frac{32 y}{3}}}^{2} + y^{2} = 100$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.1.2.1.3

Multiply ${\sqrt{- \frac{32 y}{3}}}^{2}$ by $1$ .

${\sqrt{- \frac{32 y}{3}}}^{2} + y^{2} = 100$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.1.2.1.4

Rewrite ${\sqrt{- \frac{32 y}{3}}}^{2}$ as $- \frac{32 y}{3}$ .

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Step 3.1.2.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- \frac{32 y}{3}}$ as ${(- \frac{32 y}{3})}^{\frac{1}{2}}$ .

${({(- \frac{32 y}{3})}^{\frac{1}{2}})}^{2} + y^{2} = 100$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.1.2.1.4.2

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

${(- \frac{32 y}{3})}^{\frac{1}{2} \cdot 2} + y^{2} = 100$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.1.2.1.4.3

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

${(- \frac{32 y}{3})}^{\frac{2}{2}} + y^{2} = 100$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.1.2.1.4.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.4.4.1

Cancel the common factor.

${(- \frac{32 y}{3})}^{\frac{2}{2}} + y^{2} = 100$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.1.2.1.4.4.2

Rewrite the expression.

$(- \frac{32 y}{3}) + y^{2} = 100$

$x = - \sqrt{- \frac{32 y}{3}}$

$(- \frac{32 y}{3}) + y^{2} = 100$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.1.2.1.4.5

Simplify.

$- \frac{32 y}{3} + y^{2} = 100$

$x = - \sqrt{- \frac{32 y}{3}}$

$- \frac{32 y}{3} + y^{2} = 100$

$x = - \sqrt{- \frac{32 y}{3}}$

$- \frac{32 y}{3} + y^{2} = 100$

$x = - \sqrt{- \frac{32 y}{3}}$

$- \frac{32 y}{3} + y^{2} = 100$

$x = - \sqrt{- \frac{32 y}{3}}$

$- \frac{32 y}{3} + y^{2} = 100$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2

Solve for $y$ in $- \frac{32 y}{3} + y^{2} = 100$ .

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Step 3.2.1

Multiply each term in $- \frac{32 y}{3} + y^{2} = 100$ by $3$ to eliminate the fractions.

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Step 3.2.1.1

Multiply each term in $- \frac{32 y}{3} + y^{2} = 100$ by $3$ .

$- \frac{32 y}{3} \cdot 3 + y^{2} \cdot 3 = 100 \cdot 3$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.1.2

Simplify the left side.

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Step 3.2.1.2.1

Simplify each term.

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Step 3.2.1.2.1.1

Cancel the common factor of $3$ .

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Step 3.2.1.2.1.1.1

Move the leading negative in $- \frac{32 y}{3}$ into the numerator.

$\frac{- 32 y}{3} \cdot 3 + y^{2} \cdot 3 = 100 \cdot 3$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.1.2.1.1.2

Cancel the common factor.

$\frac{- 32 y}{3} \cdot 3 + y^{2} \cdot 3 = 100 \cdot 3$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.1.2.1.1.3

Rewrite the expression.

$- 32 y + y^{2} \cdot 3 = 100 \cdot 3$

$x = - \sqrt{- \frac{32 y}{3}}$

$- 32 y + y^{2} \cdot 3 = 100 \cdot 3$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.1.2.1.2

Move $3$ to the left of $y^{2}$ .

$- 32 y + 3 y^{2} = 100 \cdot 3$

$x = - \sqrt{- \frac{32 y}{3}}$

$- 32 y + 3 y^{2} = 100 \cdot 3$

$x = - \sqrt{- \frac{32 y}{3}}$

$- 32 y + 3 y^{2} = 100 \cdot 3$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.1.3

Simplify the right side.

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Step 3.2.1.3.1

Multiply $100$ by $3$ .

$- 32 y + 3 y^{2} = 300$

$x = - \sqrt{- \frac{32 y}{3}}$

$- 32 y + 3 y^{2} = 300$

$x = - \sqrt{- \frac{32 y}{3}}$

$- 32 y + 3 y^{2} = 300$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.2

Subtract $300$ from both sides of the equation.

$- 32 y + 3 y^{2} - 300 = 0$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.3

Factor the left side of the equation.

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Step 3.2.3.1

Let $u = y$ . Substitute $u$ for all occurrences of $y$ .

$- 32 u + 3 u^{2} - 300 = 0$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.3.2

Factor by grouping.

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Step 3.2.3.2.1

Reorder terms.

$3 u^{2} - 32 u - 300 = 0$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.3.2.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 300 = - 900$ and whose sum is $b = - 32$ .

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Step 3.2.3.2.2.1

Factor $- 32$ out of $- 32 u$ .

$3 u^{2} - 32 u - 300 = 0$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.3.2.2.2

Rewrite $- 32$ as $18$ plus $- 50$

$3 u^{2} + (18 - 50) u - 300 = 0$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.3.2.2.3

Apply the distributive property.

$3 u^{2} + 18 u - 50 u - 300 = 0$

$x = - \sqrt{- \frac{32 y}{3}}$

$3 u^{2} + 18 u - 50 u - 300 = 0$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.3.2.3

Factor out the greatest common factor from each group.

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Step 3.2.3.2.3.1

Group the first two terms and the last two terms.

$(3 u^{2} + 18 u) - 50 u - 300 = 0$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.3.2.3.2

Factor out the greatest common factor (GCF) from each group.

$3 u (u + 6) - 50 (u + 6) = 0$

$x = - \sqrt{- \frac{32 y}{3}}$

$3 u (u + 6) - 50 (u + 6) = 0$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.3.2.4

Factor the polynomial by factoring out the greatest common factor, $u + 6$ .

$(u + 6) (3 u - 50) = 0$

$x = - \sqrt{- \frac{32 y}{3}}$

$(u + 6) (3 u - 50) = 0$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.3.3

Replace all occurrences of $u$ with $y$ .

$(y + 6) (3 y - 50) = 0$

$x = - \sqrt{- \frac{32 y}{3}}$

$(y + 6) (3 y - 50) = 0$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y + 6 = 0$

$3 y - 50 = 0$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.5

Set $y + 6$ equal to $0$ and solve for $y$ .

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Step 3.2.5.1

Set $y + 6$ equal to $0$ .

$y + 6 = 0$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.5.2

Subtract $6$ from both sides of the equation.

$y = - 6$

$x = - \sqrt{- \frac{32 y}{3}}$

$y = - 6$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.6

Set $3 y - 50$ equal to $0$ and solve for $y$ .

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Step 3.2.6.1

Set $3 y - 50$ equal to $0$ .

$3 y - 50 = 0$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.6.2

Solve $3 y - 50 = 0$ for $y$ .

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Step 3.2.6.2.1

Add $50$ to both sides of the equation.

$3 y = 50$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.6.2.2

Divide each term in $3 y = 50$ by $3$ and simplify.

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Step 3.2.6.2.2.1

Divide each term in $3 y = 50$ by $3$ .

$\frac{3 y}{3} = \frac{50}{3}$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.6.2.2.2

Simplify the left side.

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Step 3.2.6.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.6.2.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{50}{3}$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.6.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{50}{3}$

$x = - \sqrt{- \frac{32 y}{3}}$

$y = \frac{50}{3}$

$x = - \sqrt{- \frac{32 y}{3}}$

$y = \frac{50}{3}$

$x = - \sqrt{- \frac{32 y}{3}}$

$y = \frac{50}{3}$

$x = - \sqrt{- \frac{32 y}{3}}$

$y = \frac{50}{3}$

$x = - \sqrt{- \frac{32 y}{3}}$

$y = \frac{50}{3}$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.2.7

The final solution is all the values that make $(y + 6) (3 y - 50) = 0$ true.

$y = - 6, \frac{50}{3}$

$x = - \sqrt{- \frac{32 y}{3}}$

$y = - 6, \frac{50}{3}$

$x = - \sqrt{- \frac{32 y}{3}}$

Step 3.3

Replace all occurrences of $y$ with $- 6$ in each equation.

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Step 3.3.1

Replace all occurrences of $y$ in $x = - \sqrt{- \frac{32 y}{3}}$ with $- 6$ .

$x = - \sqrt{- \frac{32 (- 6)}{3}}$

$y = - 6$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \sqrt{- \frac{32 (- 6)}{3}}$ .

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Step 3.3.2.1.1

Reduce the expression by cancelling the common factors.

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Step 3.3.2.1.1.1

Reduce the expression $\frac{32 (- 6)}{3}$ by cancelling the common factors.

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Step 3.3.2.1.1.1.1

Factor $3$ out of $32 (- 6)$ .

$x = - \sqrt{- \frac{3 (32 \cdot (- 2))}{3}}$

$y = - 6$

Step 3.3.2.1.1.1.2

Factor $3$ out of $3$ .

$x = - \sqrt{- \frac{3 (32 \cdot (- 2))}{3 (1)}}$

$y = - 6$

Step 3.3.2.1.1.1.3

Cancel the common factor.

$x = - \sqrt{- \frac{3 (32 \cdot (- 2))}{3 \cdot 1}}$

$y = - 6$

Step 3.3.2.1.1.1.4

Rewrite the expression.

$x = - \sqrt{- \frac{32 \cdot (- 2)}{1}}$

$y = - 6$

$x = - \sqrt{- \frac{32 \cdot (- 2)}{1}}$

$y = - 6$

Step 3.3.2.1.1.2

Divide $32 \cdot (- 2)$ by $1$ .

$x = - \sqrt{- (32 \cdot (- 2))}$

$y = - 6$

$x = - \sqrt{- 1 \cdot 32 \cdot - 2}$

$y = - 6$

Step 3.3.2.1.2

Combine exponents.

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Step 3.3.2.1.2.1

Multiply $- 1$ by $32$ .

$x = - \sqrt{- 32 \cdot - 2}$

$y = - 6$

Step 3.3.2.1.2.2

Multiply $- 32$ by $- 2$ .

$x = - \sqrt{64}$

$y = - 6$

$x = - \sqrt{64}$

$y = - 6$

Step 3.3.2.1.3

Rewrite $64$ as $8^{2}$ .

$x = - \sqrt{8^{2}}$

$y = - 6$

Step 3.3.2.1.4

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

$x = - 1 \cdot 8$

$y = - 6$

Step 3.3.2.1.5

Multiply $- 1$ by $8$ .

$x = - 8$

$y = - 6$

$x = - 8$

$y = - 6$

$x = - 8$

$y = - 6$

$x = - 8$

$y = - 6$

Step 3.4

Replace all occurrences of $y$ with $\frac{50}{3}$ in each equation.

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Step 3.4.1

Replace all occurrences of $y$ in $x = - \sqrt{- \frac{32 y}{3}}$ with $\frac{50}{3}$ .

$x = - \sqrt{- \frac{32 (\frac{50}{3})}{3}}$

$y = \frac{50}{3}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- \sqrt{- \frac{32 (\frac{50}{3})}{3}}$ .

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Step 3.4.2.1.1

Combine $32$ and $\frac{50}{3}$ .

$x = - \sqrt{- \frac{\frac{32 \cdot 50}{3}}{3}}$

$y = \frac{50}{3}$

Step 3.4.2.1.2

Multiply $32$ by $50$ .

$x = - \sqrt{- \frac{\frac{1600}{3}}{3}}$

$y = \frac{50}{3}$

Step 3.4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$x = - \sqrt{- (\frac{1600}{3} \cdot \frac{1}{3})}$

$y = \frac{50}{3}$

Step 3.4.2.1.4

Multiply $\frac{1600}{3} \cdot \frac{1}{3}$ .

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Step 3.4.2.1.4.1

Multiply $\frac{1600}{3}$ by $\frac{1}{3}$ .

$x = - \sqrt{- \frac{1600}{3 \cdot 3}}$

$y = \frac{50}{3}$

Step 3.4.2.1.4.2

Multiply $3$ by $3$ .

$x = - \sqrt{- \frac{1600}{9}}$

$y = \frac{50}{3}$

$x = - \sqrt{- \frac{1600}{9}}$

$y = \frac{50}{3}$

Step 3.4.2.1.5

Rewrite $- \frac{1600}{9}$ as ${(\frac{40 i}{3})}^{2}$ .

$x = - \sqrt{{(\frac{40 i}{3})}^{2}}$

$y = \frac{50}{3}$

Step 3.4.2.1.6

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

$x = - \frac{40 i}{3}$

$y = \frac{50}{3}$

$x = - \frac{40 i}{3}$

$y = \frac{50}{3}$

$x = - \frac{40 i}{3}$

$y = \frac{50}{3}$

$x = - \frac{40 i}{3}$

$y = \frac{50}{3}$

$x = - \frac{40 i}{3}$

$y = \frac{50}{3}$

Step 4

List all of the solutions.

$x = 8, y = - 6$

$x = \frac{40 i}{3}, y = \frac{50}{3}$

$x = - 8, y = - 6$

$x = - \frac{40 i}{3}, y = \frac{50}{3}$

Step 5