Algebra Examples

$y = \sqrt{\frac{x + 1}{x - 4}}$

Step 1

Multiply the equation by $x - 4$ .

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

Step 2

Simplify the left side.

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Step 2.1

Apply the distributive property.

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

Step 3

Simplify the right side.

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Step 3.1

Simplify $(\sqrt{\frac{x + 1}{x - 4}}) (x - 4)$ .

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Step 3.1.1

Rewrite $\sqrt{\frac{x + 1}{x - 4}}$ as $\frac{\sqrt{x + 1}}{\sqrt{x - 4}}$ .

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

Step 3.1.2

Multiply $\frac{\sqrt{x + 1}}{\sqrt{x - 4}}$ by $\frac{\sqrt{x - 4}}{\sqrt{x - 4}}$ .

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

Step 3.1.3

Combine and simplify the denominator.

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Step 3.1.3.1

Multiply $\frac{\sqrt{x + 1}}{\sqrt{x - 4}}$ by $\frac{\sqrt{x - 4}}{\sqrt{x - 4}}$ .

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

Step 3.1.3.2

Raise $\sqrt{x - 4}$ to the power of $1$ .

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

Step 3.1.3.3

Raise $\sqrt{x - 4}$ to the power of $1$ .

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

Step 3.1.3.4

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

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

Step 3.1.3.5

Add $1$ and $1$ .

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

Step 3.1.3.6

Rewrite ${\sqrt{x - 4}}^{2}$ as $x - 4$ .

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Step 3.1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 4}$ as ${(x - 4)}^{\frac{1}{2}}$ .

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

Step 3.1.3.6.2

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

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

Step 3.1.3.6.3

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

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

Step 3.1.3.6.4

Cancel the common factor of $2$ .

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Step 3.1.3.6.4.1

Cancel the common factor.

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

Step 3.1.3.6.4.2

Rewrite the expression.

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

Step 3.1.3.6.5

Simplify.

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

Step 3.1.4

Combine using the product rule for radicals.

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

Step 3.1.5

Cancel the common factor of $x - 4$ .

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Step 3.1.5.1

Cancel the common factor.

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

Step 3.1.5.2

Rewrite the expression.

$x y - 4 y = \sqrt{(x + 1) (x - 4)}$

Step 4

Solve for $x$ .

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Step 4.1

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

$\sqrt{(x + 1) (x - 4)} = x y - 4 y$

Step 4.2

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 4.3

Simplify each side of the equation.

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Step 4.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(x + 1) (x - 4)}$ as ${((x + 1) (x - 4))}^{\frac{1}{2}}$ .

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

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

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Step 4.3.2.1.1

Multiply the exponents in ${({((x + 1) (x - 4))}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.2.1.1.1

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

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

Step 4.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.3.2.1.1.2.1

Cancel the common factor.

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

Step 4.3.2.1.1.2.2

Rewrite the expression.

${((x + 1) (x - 4))}^{1} = {(x y - 4 y)}^{2}$

Step 4.3.2.1.2

Expand $(x + 1) (x - 4)$ using the FOIL Method.

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Step 4.3.2.1.2.1

Apply the distributive property.

${(x (x - 4) + 1 (x - 4))}^{1} = {(x y - 4 y)}^{2}$

Step 4.3.2.1.2.2

Apply the distributive property.

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

Step 4.3.2.1.2.3

Apply the distributive property.

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

Step 4.3.2.1.3

Simplify and combine like terms.

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Step 4.3.2.1.3.1

Simplify each term.

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Step 4.3.2.1.3.1.1

Multiply $x$ by $x$ .

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

Step 4.3.2.1.3.1.2

Move $- 4$ to the left of $x$ .

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

Step 4.3.2.1.3.1.3

Multiply $x$ by $1$ .

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

Step 4.3.2.1.3.1.4

Multiply $- 4$ by $1$ .

${(x^{2} - 4 x + x - 4)}^{1} = {(x y - 4 y)}^{2}$

Step 4.3.2.1.3.2

Add $- 4 x$ and $x$ .

${(x^{2} - 3 x - 4)}^{1} = {(x y - 4 y)}^{2}$

Step 4.3.2.1.4

Simplify.

$x^{2} - 3 x - 4 = {(x y - 4 y)}^{2}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

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

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Step 4.3.3.1.1

Rewrite ${(x y - 4 y)}^{2}$ as $(x y - 4 y) (x y - 4 y)$ .

$x^{2} - 3 x - 4 = (x y - 4 y) (x y - 4 y)$

Step 4.3.3.1.2

Expand $(x y - 4 y) (x y - 4 y)$ using the FOIL Method.

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Step 4.3.3.1.2.1

Apply the distributive property.

$x^{2} - 3 x - 4 = x y (x y - 4 y) - 4 y (x y - 4 y)$

Step 4.3.3.1.2.2

Apply the distributive property.

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

Step 4.3.3.1.2.3

Apply the distributive property.

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

Step 4.3.3.1.3

Simplify and combine like terms.

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Step 4.3.3.1.3.1

Simplify each term.

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Step 4.3.3.1.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.3.3.1.3.1.1.1

Move $x$ .

$x^{2} - 3 x - 4 = x \cdot x y \cdot y + x y (- 4 y) - 4 y (x y) - 4 y (- 4 y)$

Step 4.3.3.1.3.1.1.2

Multiply $x$ by $x$ .

$x^{2} - 3 x - 4 = x^{2} y \cdot y + x y (- 4 y) - 4 y (x y) - 4 y (- 4 y)$

Step 4.3.3.1.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 4.3.3.1.3.1.2.1

Move $y$ .

$x^{2} - 3 x - 4 = x^{2} (y \cdot y) + x y (- 4 y) - 4 y (x y) - 4 y (- 4 y)$

Step 4.3.3.1.3.1.2.2

Multiply $y$ by $y$ .

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

Step 4.3.3.1.3.1.3

Rewrite using the commutative property of multiplication.

$x^{2} - 3 x - 4 = x^{2} y^{2} - 4 x y \cdot y - 4 y (x y) - 4 y (- 4 y)$

Step 4.3.3.1.3.1.4

Multiply $y$ by $y$ by adding the exponents.

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Step 4.3.3.1.3.1.4.1

Move $y$ .

$x^{2} - 3 x - 4 = x^{2} y^{2} - 4 x (y \cdot y) - 4 y (x y) - 4 y (- 4 y)$

Step 4.3.3.1.3.1.4.2

Multiply $y$ by $y$ .

$x^{2} - 3 x - 4 = x^{2} y^{2} - 4 x y^{2} - 4 y (x y) - 4 y (- 4 y)$

Step 4.3.3.1.3.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 4.3.3.1.3.1.5.1

Move $y$ .

$x^{2} - 3 x - 4 = x^{2} y^{2} - 4 x y^{2} - 4 (y \cdot y) x - 4 y (- 4 y)$

Step 4.3.3.1.3.1.5.2

Multiply $y$ by $y$ .

$x^{2} - 3 x - 4 = x^{2} y^{2} - 4 x y^{2} - 4 y^{2} x - 4 y (- 4 y)$

Step 4.3.3.1.3.1.6

Rewrite using the commutative property of multiplication.

$x^{2} - 3 x - 4 = x^{2} y^{2} - 4 x y^{2} - 4 y^{2} x - 4 \cdot - 4 y \cdot y$

Step 4.3.3.1.3.1.7

Multiply $y$ by $y$ by adding the exponents.

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Step 4.3.3.1.3.1.7.1

Move $y$ .

$x^{2} - 3 x - 4 = x^{2} y^{2} - 4 x y^{2} - 4 y^{2} x - 4 \cdot - 4 (y \cdot y)$

Step 4.3.3.1.3.1.7.2

Multiply $y$ by $y$ .

$x^{2} - 3 x - 4 = x^{2} y^{2} - 4 x y^{2} - 4 y^{2} x - 4 \cdot - 4 y^{2}$

Step 4.3.3.1.3.1.8

Multiply $- 4$ by $- 4$ .

$x^{2} - 3 x - 4 = x^{2} y^{2} - 4 x y^{2} - 4 y^{2} x + 16 y^{2}$

Step 4.3.3.1.3.2

Subtract $4 y^{2} x$ from $- 4 x y^{2}$ .

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Step 4.3.3.1.3.2.1

Move $y^{2}$ .

$x^{2} - 3 x - 4 = x^{2} y^{2} - 4 x y^{2} - 4 x y^{2} + 16 y^{2}$

Step 4.3.3.1.3.2.2

Subtract $4 x y^{2}$ from $- 4 x y^{2}$ .

$x^{2} - 3 x - 4 = x^{2} y^{2} - 8 x y^{2} + 16 y^{2}$

Step 4.4

Solve for $x$ .

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Step 4.4.1

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

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Step 4.4.1.1

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

$x^{2} - 3 x - 4 - x^{2} y^{2} = - 8 x y^{2} + 16 y^{2}$

Step 4.4.1.2

Add $8 x y^{2}$ to both sides of the equation.

$x^{2} - 3 x - 4 - x^{2} y^{2} + 8 x y^{2} = 16 y^{2}$

Step 4.4.2

Subtract $16 y^{2}$ from both sides of the equation.

$x^{2} - 3 x - 4 - x^{2} y^{2} + 8 x y^{2} - 16 y^{2} = 0$

Step 4.4.3

Use the quadratic formula to find the solutions.

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

Step 4.4.4

Substitute the values $a = 1 - y^{2}$ , $b = - 3 + 8 y^{2}$ , and $c = - 4 - 16 y^{2}$ into the quadratic formula and solve for $x$ .

$\frac{- (- 3 + 8 y^{2}) \pm \sqrt{{(- 3 + 8 y^{2})}^{2} - 4 \cdot ((1 - y^{2}) \cdot (- 4 - 16 y^{2}))}}{2 (1 - y^{2})}$

Step 4.4.5

Simplify.

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Step 4.4.5.1

Simplify the numerator.

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Step 4.4.5.1.1

Apply the distributive property.

$x = \frac{3 - (8 y^{2}) \pm \sqrt{{(- 3 + 8 y^{2})}^{2} - 4 \cdot (1 - y^{2}) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.2

Multiply $- 1$ by $- 3$ .

$x = \frac{3 - (8 y^{2}) \pm \sqrt{{(- 3 + 8 y^{2})}^{2} - 4 \cdot (1 - y^{2}) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.3

Multiply $8$ by $- 1$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{{(- 3 + 8 y^{2})}^{2} - 4 \cdot (1 - y^{2}) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.4

Rewrite ${(- 3 + 8 y^{2})}^{2}$ as $(- 3 + 8 y^{2}) (- 3 + 8 y^{2})$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{(- 3 + 8 y^{2}) (- 3 + 8 y^{2}) - 4 \cdot (1 - y^{2}) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.5

Expand $(- 3 + 8 y^{2}) (- 3 + 8 y^{2})$ using the FOIL Method.

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Step 4.4.5.1.5.1

Apply the distributive property.

$x = \frac{3 - 8 y^{2} \pm \sqrt{- 3 (- 3 + 8 y^{2}) + 8 y^{2} (- 3 + 8 y^{2}) - 4 \cdot (1 - y^{2}) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.5.2

Apply the distributive property.

$x = \frac{3 - 8 y^{2} \pm \sqrt{- 3 \cdot - 3 - 3 (8 y^{2}) + 8 y^{2} (- 3 + 8 y^{2}) - 4 \cdot (1 - y^{2}) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.5.3

Apply the distributive property.

$x = \frac{3 - 8 y^{2} \pm \sqrt{- 3 \cdot - 3 - 3 (8 y^{2}) + 8 y^{2} \cdot - 3 + 8 y^{2} (8 y^{2}) - 4 \cdot (1 - y^{2}) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.6

Simplify and combine like terms.

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Step 4.4.5.1.6.1

Simplify each term.

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Step 4.4.5.1.6.1.1

Multiply $- 3$ by $- 3$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 3 (8 y^{2}) + 8 y^{2} \cdot - 3 + 8 y^{2} (8 y^{2}) - 4 \cdot (1 - y^{2}) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.6.1.2

Multiply $8$ by $- 3$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 24 y^{2} + 8 y^{2} \cdot - 3 + 8 y^{2} (8 y^{2}) - 4 \cdot (1 - y^{2}) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.6.1.3

Multiply $- 3$ by $8$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 24 y^{2} - 24 y^{2} + 8 y^{2} (8 y^{2}) - 4 \cdot (1 - y^{2}) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.6.1.4

Rewrite using the commutative property of multiplication.

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 24 y^{2} - 24 y^{2} + 8 \cdot (8 y^{2} y^{2}) - 4 \cdot (1 - y^{2}) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.6.1.5

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 4.4.5.1.6.1.5.1

Move $y^{2}$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 24 y^{2} - 24 y^{2} + 8 \cdot (8 (y^{2} y^{2})) - 4 \cdot (1 - y^{2}) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.6.1.5.2

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

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 24 y^{2} - 24 y^{2} + 8 \cdot (8 y^{2 + 2}) - 4 \cdot (1 - y^{2}) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.6.1.5.3

Add $2$ and $2$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 24 y^{2} - 24 y^{2} + 8 \cdot (8 y^{4}) - 4 \cdot (1 - y^{2}) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.6.1.6

Multiply $8$ by $8$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 24 y^{2} - 24 y^{2} + 64 y^{4} - 4 \cdot (1 - y^{2}) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.6.2

Subtract $24 y^{2}$ from $- 24 y^{2}$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 48 y^{2} + 64 y^{4} - 4 \cdot (1 - y^{2}) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.7

Apply the distributive property.

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 48 y^{2} + 64 y^{4} + (- 4 \cdot 1 - 4 (- y^{2})) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.8

Multiply $- 4$ by $1$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 48 y^{2} + 64 y^{4} + (- 4 - 4 (- y^{2})) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.9

Multiply $- 1$ by $- 4$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 48 y^{2} + 64 y^{4} + (- 4 + 4 y^{2}) \cdot (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.10

Expand $(- 4 + 4 y^{2}) (- 4 - 16 y^{2})$ using the FOIL Method.

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Step 4.4.5.1.10.1

Apply the distributive property.

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 48 y^{2} + 64 y^{4} - 4 (- 4 - 16 y^{2}) + 4 y^{2} (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.10.2

Apply the distributive property.

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 48 y^{2} + 64 y^{4} - 4 \cdot - 4 - 4 (- 16 y^{2}) + 4 y^{2} (- 4 - 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.10.3

Apply the distributive property.

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 48 y^{2} + 64 y^{4} - 4 \cdot - 4 - 4 (- 16 y^{2}) + 4 y^{2} \cdot - 4 + 4 y^{2} (- 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.11

Simplify and combine like terms.

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Step 4.4.5.1.11.1

Simplify each term.

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Step 4.4.5.1.11.1.1

Multiply $- 4$ by $- 4$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 48 y^{2} + 64 y^{4} + 16 - 4 (- 16 y^{2}) + 4 y^{2} \cdot - 4 + 4 y^{2} (- 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.11.1.2

Multiply $- 16$ by $- 4$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 48 y^{2} + 64 y^{4} + 16 + 64 y^{2} + 4 y^{2} \cdot - 4 + 4 y^{2} (- 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.11.1.3

Multiply $- 4$ by $4$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 48 y^{2} + 64 y^{4} + 16 + 64 y^{2} - 16 y^{2} + 4 y^{2} (- 16 y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.11.1.4

Rewrite using the commutative property of multiplication.

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 48 y^{2} + 64 y^{4} + 16 + 64 y^{2} - 16 y^{2} + 4 \cdot (- 16 y^{2} y^{2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.11.1.5

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 4.4.5.1.11.1.5.1

Move $y^{2}$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 48 y^{2} + 64 y^{4} + 16 + 64 y^{2} - 16 y^{2} + 4 \cdot (- 16 (y^{2} y^{2}))}}{2 (1 - y^{2})}$

Step 4.4.5.1.11.1.5.2

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

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 48 y^{2} + 64 y^{4} + 16 + 64 y^{2} - 16 y^{2} + 4 \cdot (- 16 y^{2 + 2})}}{2 (1 - y^{2})}$

Step 4.4.5.1.11.1.5.3

Add $2$ and $2$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 48 y^{2} + 64 y^{4} + 16 + 64 y^{2} - 16 y^{2} + 4 \cdot (- 16 y^{4})}}{2 (1 - y^{2})}$

Step 4.4.5.1.11.1.6

Multiply $4$ by $- 16$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 48 y^{2} + 64 y^{4} + 16 + 64 y^{2} - 16 y^{2} - 64 y^{4}}}{2 (1 - y^{2})}$

Step 4.4.5.1.11.2

Subtract $16 y^{2}$ from $64 y^{2}$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{9 - 48 y^{2} + 64 y^{4} + 16 + 48 y^{2} - 64 y^{4}}}{2 (1 - y^{2})}$

Step 4.4.5.1.12

Add $9$ and $16$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{- 48 y^{2} + 64 y^{4} + 25 + 48 y^{2} - 64 y^{4}}}{2 (1 - y^{2})}$

Step 4.4.5.1.13

Add $- 48 y^{2}$ and $48 y^{2}$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{64 y^{4} + 25 + 0 - 64 y^{4}}}{2 (1 - y^{2})}$

Step 4.4.5.1.14

Add $64 y^{4}$ and $0$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{64 y^{4} + 25 - 64 y^{4}}}{2 (1 - y^{2})}$

Step 4.4.5.1.15

Subtract $64 y^{4}$ from $64 y^{4}$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{0 + 25}}{2 (1 - y^{2})}$

Step 4.4.5.1.16

Add $0$ and $25$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{25}}{2 (1 - y^{2})}$

Step 4.4.5.1.17

Rewrite $25$ as $5^{2}$ .

$x = \frac{3 - 8 y^{2} \pm \sqrt{5^{2}}}{2 (1 - y^{2})}$

Step 4.4.5.1.18

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

$x = \frac{3 - 8 y^{2} \pm 5}{2 (1 - y^{2})}$

Step 4.4.5.2

Simplify the denominator.

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Step 4.4.5.2.1

Rewrite $1$ as $1^{2}$ .

$x = \frac{3 - 8 y^{2} \pm 5}{2 (1^{2} - y^{2})}$

Step 4.4.5.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = y$ .

$x = \frac{3 - 8 y^{2} \pm 5}{2 (1 + y) (1 - y)}$

Step 4.4.6

The final answer is the combination of both solutions.

$x = 4$

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

$x = 4$

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

$x = 4$

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