Precalculus Examples

$f (x) = \sqrt{\frac{x + 3}{x - 2}}$

Step 1

Write $f (x) = \sqrt{\frac{x + 3}{x - 2}}$ as an equation.

$y = \sqrt{\frac{x + 3}{x - 2}}$

Step 2

Interchange the variables.

$x = \sqrt{\frac{y + 3}{y - 2}}$

Step 3

Solve for $y$ .

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Step 3.1

Multiply the equation by $y - 2$ .

$(y - 2) (x) = (\sqrt{\frac{y + 3}{y - 2}}) (y - 2)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

$y x - 2 x = (\sqrt{\frac{y + 3}{y - 2}}) (y - 2)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify $(\sqrt{\frac{y + 3}{y - 2}}) (y - 2)$ .

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Step 3.3.1.1

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

$y x - 2 x = \frac{\sqrt{y + 3}}{\sqrt{y - 2}} (y - 2)$

Step 3.3.1.2

Multiply $\frac{\sqrt{y + 3}}{\sqrt{y - 2}}$ by $\frac{\sqrt{y - 2}}{\sqrt{y - 2}}$ .

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

Step 3.3.1.3

Combine and simplify the denominator.

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Step 3.3.1.3.1

Multiply $\frac{\sqrt{y + 3}}{\sqrt{y - 2}}$ by $\frac{\sqrt{y - 2}}{\sqrt{y - 2}}$ .

$y x - 2 x = \frac{\sqrt{y + 3} \sqrt{y - 2}}{\sqrt{y - 2} \sqrt{y - 2}} (y - 2)$

Step 3.3.1.3.2

Raise $\sqrt{y - 2}$ to the power of $1$ .

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

Step 3.3.1.3.3

Raise $\sqrt{y - 2}$ to the power of $1$ .

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

Step 3.3.1.3.4

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

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

Step 3.3.1.3.5

Add $1$ and $1$ .

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

Step 3.3.1.3.6

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

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Step 3.3.1.3.6.1

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

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

Step 3.3.1.3.6.2

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

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

Step 3.3.1.3.6.3

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

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

Step 3.3.1.3.6.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.6.4.1

Cancel the common factor.

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

Step 3.3.1.3.6.4.2

Rewrite the expression.

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

Step 3.3.1.3.6.5

Simplify.

$y x - 2 x = \frac{\sqrt{y + 3} \sqrt{y - 2}}{y - 2} (y - 2)$

Step 3.3.1.4

Combine using the product rule for radicals.

$y x - 2 x = \frac{\sqrt{(y + 3) (y - 2)}}{y - 2} (y - 2)$

Step 3.3.1.5

Cancel the common factor of $y - 2$ .

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Step 3.3.1.5.1

Cancel the common factor.

$y x - 2 x = \frac{\sqrt{(y + 3) (y - 2)}}{y - 2} (y - 2)$

Step 3.3.1.5.2

Rewrite the expression.

$y x - 2 x = \sqrt{(y + 3) (y - 2)}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

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

$\sqrt{(y + 3) (y - 2)} = y x - 2 x$

Step 3.4.2

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

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

Step 3.4.3

Simplify each side of the equation.

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Step 3.4.3.1

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

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

Step 3.4.3.2

Simplify the left side.

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Step 3.4.3.2.1

Simplify ${({((y + 3) (y - 2))}^{\frac{1}{2}})}^{2}$ .

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Step 3.4.3.2.1.1

Multiply the exponents in ${({((y + 3) (y - 2))}^{\frac{1}{2}})}^{2}$ .

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Step 3.4.3.2.1.1.1

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

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

Step 3.4.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.4.3.2.1.1.2.1

Cancel the common factor.

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

Step 3.4.3.2.1.1.2.2

Rewrite the expression.

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

Step 3.4.3.2.1.2

Expand $(y + 3) (y - 2)$ using the FOIL Method.

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Step 3.4.3.2.1.2.1

Apply the distributive property.

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

Step 3.4.3.2.1.2.2

Apply the distributive property.

${(y \cdot y + y \cdot - 2 + 3 (y - 2))}^{1} = {(y x - 2 x)}^{2}$

Step 3.4.3.2.1.2.3

Apply the distributive property.

${(y \cdot y + y \cdot - 2 + 3 y + 3 \cdot - 2)}^{1} = {(y x - 2 x)}^{2}$

Step 3.4.3.2.1.3

Simplify and combine like terms.

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Step 3.4.3.2.1.3.1

Simplify each term.

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Step 3.4.3.2.1.3.1.1

Multiply $y$ by $y$ .

${(y^{2} + y \cdot - 2 + 3 y + 3 \cdot - 2)}^{1} = {(y x - 2 x)}^{2}$

Step 3.4.3.2.1.3.1.2

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

${(y^{2} - 2 \cdot y + 3 y + 3 \cdot - 2)}^{1} = {(y x - 2 x)}^{2}$

Step 3.4.3.2.1.3.1.3

Multiply $3$ by $- 2$ .

${(y^{2} - 2 y + 3 y - 6)}^{1} = {(y x - 2 x)}^{2}$

Step 3.4.3.2.1.3.2

Add $- 2 y$ and $3 y$ .

${(y^{2} + y - 6)}^{1} = {(y x - 2 x)}^{2}$

Step 3.4.3.2.1.4

Simplify.

$y^{2} + y - 6 = {(y x - 2 x)}^{2}$

Step 3.4.3.3

Simplify the right side.

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Step 3.4.3.3.1

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

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Step 3.4.3.3.1.1

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

$y^{2} + y - 6 = (y x - 2 x) (y x - 2 x)$

Step 3.4.3.3.1.2

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

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Step 3.4.3.3.1.2.1

Apply the distributive property.

$y^{2} + y - 6 = y x (y x - 2 x) - 2 x (y x - 2 x)$

Step 3.4.3.3.1.2.2

Apply the distributive property.

$y^{2} + y - 6 = y x (y x) + y x (- 2 x) - 2 x (y x - 2 x)$

Step 3.4.3.3.1.2.3

Apply the distributive property.

$y^{2} + y - 6 = y x (y x) + y x (- 2 x) - 2 x (y x) - 2 x (- 2 x)$

Step 3.4.3.3.1.3

Simplify and combine like terms.

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Step 3.4.3.3.1.3.1

Simplify each term.

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Step 3.4.3.3.1.3.1.1

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

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Step 3.4.3.3.1.3.1.1.1

Move $y$ .

$y^{2} + y - 6 = y \cdot y x \cdot x + y x (- 2 x) - 2 x (y x) - 2 x (- 2 x)$

Step 3.4.3.3.1.3.1.1.2

Multiply $y$ by $y$ .

$y^{2} + y - 6 = y^{2} x \cdot x + y x (- 2 x) - 2 x (y x) - 2 x (- 2 x)$

Step 3.4.3.3.1.3.1.2

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

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Step 3.4.3.3.1.3.1.2.1

Move $x$ .

$y^{2} + y - 6 = y^{2} (x \cdot x) + y x (- 2 x) - 2 x (y x) - 2 x (- 2 x)$

Step 3.4.3.3.1.3.1.2.2

Multiply $x$ by $x$ .

$y^{2} + y - 6 = y^{2} x^{2} + y x (- 2 x) - 2 x (y x) - 2 x (- 2 x)$

Step 3.4.3.3.1.3.1.3

Rewrite using the commutative property of multiplication.

$y^{2} + y - 6 = y^{2} x^{2} - 2 y x \cdot x - 2 x (y x) - 2 x (- 2 x)$

Step 3.4.3.3.1.3.1.4

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

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Step 3.4.3.3.1.3.1.4.1

Move $x$ .

$y^{2} + y - 6 = y^{2} x^{2} - 2 y (x \cdot x) - 2 x (y x) - 2 x (- 2 x)$

Step 3.4.3.3.1.3.1.4.2

Multiply $x$ by $x$ .

$y^{2} + y - 6 = y^{2} x^{2} - 2 y x^{2} - 2 x (y x) - 2 x (- 2 x)$

Step 3.4.3.3.1.3.1.5

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

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Step 3.4.3.3.1.3.1.5.1

Move $x$ .

$y^{2} + y - 6 = y^{2} x^{2} - 2 y x^{2} - 2 (x \cdot x) y - 2 x (- 2 x)$

Step 3.4.3.3.1.3.1.5.2

Multiply $x$ by $x$ .

$y^{2} + y - 6 = y^{2} x^{2} - 2 y x^{2} - 2 x^{2} y - 2 x (- 2 x)$

Step 3.4.3.3.1.3.1.6

Rewrite using the commutative property of multiplication.

$y^{2} + y - 6 = y^{2} x^{2} - 2 y x^{2} - 2 x^{2} y - 2 \cdot - 2 x \cdot x$

Step 3.4.3.3.1.3.1.7

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

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Step 3.4.3.3.1.3.1.7.1

Move $x$ .

$y^{2} + y - 6 = y^{2} x^{2} - 2 y x^{2} - 2 x^{2} y - 2 \cdot - 2 (x \cdot x)$

Step 3.4.3.3.1.3.1.7.2

Multiply $x$ by $x$ .

$y^{2} + y - 6 = y^{2} x^{2} - 2 y x^{2} - 2 x^{2} y - 2 \cdot - 2 x^{2}$

Step 3.4.3.3.1.3.1.8

Multiply $- 2$ by $- 2$ .

$y^{2} + y - 6 = y^{2} x^{2} - 2 y x^{2} - 2 x^{2} y + 4 x^{2}$

Step 3.4.3.3.1.3.2

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

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Step 3.4.3.3.1.3.2.1

Move $y$ .

$y^{2} + y - 6 = y^{2} x^{2} - 2 x^{2} y - 2 x^{2} y + 4 x^{2}$

Step 3.4.3.3.1.3.2.2

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

$y^{2} + y - 6 = y^{2} x^{2} - 4 x^{2} y + 4 x^{2}$

Step 3.4.4

Solve for $y$ .

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Step 3.4.4.1

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

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Step 3.4.4.1.1

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

$y^{2} + y - 6 - y^{2} x^{2} = - 4 x^{2} y + 4 x^{2}$

Step 3.4.4.1.2

Add $4 x^{2} y$ to both sides of the equation.

$y^{2} + y - 6 - y^{2} x^{2} + 4 x^{2} y = 4 x^{2}$

Step 3.4.4.2

Subtract $4 x^{2}$ from both sides of the equation.

$y^{2} + y - 6 - y^{2} x^{2} + 4 x^{2} y - 4 x^{2} = 0$

Step 3.4.4.3

Use the quadratic formula to find the solutions.

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

Step 3.4.4.4

Substitute the values $a = 1 - x^{2}$ , $b = 1 + 4 x^{2}$ , and $c = - 6 - 4 x^{2}$ into the quadratic formula and solve for $y$ .

$\frac{- (1 + 4 x^{2}) \pm \sqrt{{(1 + 4 x^{2})}^{2} - 4 \cdot ((1 - x^{2}) \cdot (- 6 - 4 x^{2}))}}{2 (1 - x^{2})}$

Step 3.4.4.5

Simplify.

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Step 3.4.4.5.1

Simplify the numerator.

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Step 3.4.4.5.1.1

Apply the distributive property.

$y = \frac{- 1 \cdot 1 - (4 x^{2}) \pm \sqrt{{(1 + 4 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.2

Multiply $- 1$ by $1$ .

$y = \frac{- 1 - (4 x^{2}) \pm \sqrt{{(1 + 4 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.3

Multiply $4$ by $- 1$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{{(1 + 4 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.4

Rewrite ${(1 + 4 x^{2})}^{2}$ as $(1 + 4 x^{2}) (1 + 4 x^{2})$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{(1 + 4 x^{2}) (1 + 4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.5

Expand $(1 + 4 x^{2}) (1 + 4 x^{2})$ using the FOIL Method.

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Step 3.4.4.5.1.5.1

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 (1 + 4 x^{2}) + 4 x^{2} (1 + 4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.5.2

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 \cdot 1 + 1 (4 x^{2}) + 4 x^{2} (1 + 4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.5.3

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 \cdot 1 + 1 (4 x^{2}) + 4 x^{2} \cdot 1 + 4 x^{2} (4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.6

Simplify and combine like terms.

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Step 3.4.4.5.1.6.1

Simplify each term.

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Step 3.4.4.5.1.6.1.1

Multiply $1$ by $1$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 1 (4 x^{2}) + 4 x^{2} \cdot 1 + 4 x^{2} (4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.6.1.2

Multiply $4 x^{2}$ by $1$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} \cdot 1 + 4 x^{2} (4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.6.1.3

Multiply $4$ by $1$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 4 x^{2} (4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.6.1.4

Rewrite using the commutative property of multiplication.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 4 \cdot (4 x^{2} x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.6.1.5

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

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Step 3.4.4.5.1.6.1.5.1

Move $x^{2}$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 4 \cdot (4 (x^{2} x^{2})) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.6.1.5.2

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

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 4 \cdot (4 x^{2 + 2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.6.1.5.3

Add $2$ and $2$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 4 \cdot (4 x^{4}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.6.1.6

Multiply $4$ by $4$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 16 x^{4} - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.6.2

Add $4 x^{2}$ and $4 x^{2}$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.7

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} + (- 4 \cdot 1 - 4 (- x^{2})) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.8

Multiply $- 4$ by $1$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} + (- 4 - 4 (- x^{2})) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.9

Multiply $- 1$ by $- 4$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} + (- 4 + 4 x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.10

Expand $(- 4 + 4 x^{2}) (- 6 - 4 x^{2})$ using the FOIL Method.

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Step 3.4.4.5.1.10.1

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} - 4 (- 6 - 4 x^{2}) + 4 x^{2} (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.10.2

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} - 4 \cdot - 6 - 4 (- 4 x^{2}) + 4 x^{2} (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.10.3

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} - 4 \cdot - 6 - 4 (- 4 x^{2}) + 4 x^{2} \cdot - 6 + 4 x^{2} (- 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.11

Simplify and combine like terms.

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Step 3.4.4.5.1.11.1

Simplify each term.

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Step 3.4.4.5.1.11.1.1

Multiply $- 4$ by $- 6$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} + 24 - 4 (- 4 x^{2}) + 4 x^{2} \cdot - 6 + 4 x^{2} (- 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.11.1.2

Multiply $- 4$ by $- 4$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} + 24 + 16 x^{2} + 4 x^{2} \cdot - 6 + 4 x^{2} (- 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.11.1.3

Multiply $- 6$ by $4$ .

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

Step 3.4.4.5.1.11.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.4.4.5.1.11.1.5

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

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Step 3.4.4.5.1.11.1.5.1

Move $x^{2}$ .

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

Step 3.4.4.5.1.11.1.5.2

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

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

Step 3.4.4.5.1.11.1.5.3

Add $2$ and $2$ .

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

Step 3.4.4.5.1.11.1.6

Multiply $4$ by $- 4$ .

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

Step 3.4.4.5.1.11.2

Subtract $24 x^{2}$ from $16 x^{2}$ .

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

Step 3.4.4.5.1.12

Add $1$ and $24$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{8 x^{2} + 16 x^{4} + 25 - 8 x^{2} - 16 x^{4}}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.13

Subtract $8 x^{2}$ from $8 x^{2}$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{16 x^{4} + 25 + 0 - 16 x^{4}}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.14

Add $16 x^{4}$ and $0$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{16 x^{4} + 25 - 16 x^{4}}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.15

Subtract $16 x^{4}$ from $16 x^{4}$ .

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

Step 3.4.4.5.1.16

Add $0$ and $25$ .

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

Step 3.4.4.5.1.17

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

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{5^{2}}}{2 (1 - x^{2})}$

Step 3.4.4.5.1.18

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

$y = \frac{- 1 - 4 x^{2} \pm 5}{2 (1 - x^{2})}$

Step 3.4.4.5.2

Simplify the denominator.

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Step 3.4.4.5.2.1

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

$y = \frac{- 1 - 4 x^{2} \pm 5}{2 (1^{2} - x^{2})}$

Step 3.4.4.5.2.2

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 $b = x$ .

$y = \frac{- 1 - 4 x^{2} \pm 5}{2 (1 + x) (1 - x)}$

Step 3.4.4.6

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

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Step 3.4.4.6.1

Simplify the numerator.

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Step 3.4.4.6.1.1

Apply the distributive property.

$y = \frac{- 1 \cdot 1 - (4 x^{2}) \pm \sqrt{{(1 + 4 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.2

Multiply $- 1$ by $1$ .

$y = \frac{- 1 - (4 x^{2}) \pm \sqrt{{(1 + 4 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.3

Multiply $4$ by $- 1$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{{(1 + 4 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.4

Rewrite ${(1 + 4 x^{2})}^{2}$ as $(1 + 4 x^{2}) (1 + 4 x^{2})$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{(1 + 4 x^{2}) (1 + 4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.5

Expand $(1 + 4 x^{2}) (1 + 4 x^{2})$ using the FOIL Method.

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Step 3.4.4.6.1.5.1

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 (1 + 4 x^{2}) + 4 x^{2} (1 + 4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.5.2

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 \cdot 1 + 1 (4 x^{2}) + 4 x^{2} (1 + 4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.5.3

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 \cdot 1 + 1 (4 x^{2}) + 4 x^{2} \cdot 1 + 4 x^{2} (4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.6

Simplify and combine like terms.

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Step 3.4.4.6.1.6.1

Simplify each term.

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Step 3.4.4.6.1.6.1.1

Multiply $1$ by $1$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 1 (4 x^{2}) + 4 x^{2} \cdot 1 + 4 x^{2} (4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.6.1.2

Multiply $4 x^{2}$ by $1$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} \cdot 1 + 4 x^{2} (4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.6.1.3

Multiply $4$ by $1$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 4 x^{2} (4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.6.1.4

Rewrite using the commutative property of multiplication.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 4 \cdot (4 x^{2} x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.6.1.5

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

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Step 3.4.4.6.1.6.1.5.1

Move $x^{2}$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 4 \cdot (4 (x^{2} x^{2})) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.6.1.5.2

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

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 4 \cdot (4 x^{2 + 2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.6.1.5.3

Add $2$ and $2$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 4 \cdot (4 x^{4}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.6.1.6

Multiply $4$ by $4$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 16 x^{4} - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.6.2

Add $4 x^{2}$ and $4 x^{2}$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.7

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} + (- 4 \cdot 1 - 4 (- x^{2})) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.8

Multiply $- 4$ by $1$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} + (- 4 - 4 (- x^{2})) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.9

Multiply $- 1$ by $- 4$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} + (- 4 + 4 x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.10

Expand $(- 4 + 4 x^{2}) (- 6 - 4 x^{2})$ using the FOIL Method.

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Step 3.4.4.6.1.10.1

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} - 4 (- 6 - 4 x^{2}) + 4 x^{2} (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.10.2

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} - 4 \cdot - 6 - 4 (- 4 x^{2}) + 4 x^{2} (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.10.3

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} - 4 \cdot - 6 - 4 (- 4 x^{2}) + 4 x^{2} \cdot - 6 + 4 x^{2} (- 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.11

Simplify and combine like terms.

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Step 3.4.4.6.1.11.1

Simplify each term.

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Step 3.4.4.6.1.11.1.1

Multiply $- 4$ by $- 6$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} + 24 - 4 (- 4 x^{2}) + 4 x^{2} \cdot - 6 + 4 x^{2} (- 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.11.1.2

Multiply $- 4$ by $- 4$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} + 24 + 16 x^{2} + 4 x^{2} \cdot - 6 + 4 x^{2} (- 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.11.1.3

Multiply $- 6$ by $4$ .

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

Step 3.4.4.6.1.11.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.4.4.6.1.11.1.5

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

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Step 3.4.4.6.1.11.1.5.1

Move $x^{2}$ .

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

Step 3.4.4.6.1.11.1.5.2

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

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

Step 3.4.4.6.1.11.1.5.3

Add $2$ and $2$ .

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

Step 3.4.4.6.1.11.1.6

Multiply $4$ by $- 4$ .

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

Step 3.4.4.6.1.11.2

Subtract $24 x^{2}$ from $16 x^{2}$ .

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

Step 3.4.4.6.1.12

Add $1$ and $24$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{8 x^{2} + 16 x^{4} + 25 - 8 x^{2} - 16 x^{4}}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.13

Subtract $8 x^{2}$ from $8 x^{2}$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{16 x^{4} + 25 + 0 - 16 x^{4}}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.14

Add $16 x^{4}$ and $0$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{16 x^{4} + 25 - 16 x^{4}}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.15

Subtract $16 x^{4}$ from $16 x^{4}$ .

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

Step 3.4.4.6.1.16

Add $0$ and $25$ .

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

Step 3.4.4.6.1.17

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

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{5^{2}}}{2 (1 - x^{2})}$

Step 3.4.4.6.1.18

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

$y = \frac{- 1 - 4 x^{2} \pm 5}{2 (1 - x^{2})}$

Step 3.4.4.6.2

Simplify the denominator.

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Step 3.4.4.6.2.1

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

$y = \frac{- 1 - 4 x^{2} \pm 5}{2 (1^{2} - x^{2})}$

Step 3.4.4.6.2.2

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 $b = x$ .

$y = \frac{- 1 - 4 x^{2} \pm 5}{2 (1 + x) (1 - x)}$

Step 3.4.4.6.3

Change the $\pm$ to $+$ .

$y = \frac{- 1 - 4 x^{2} + 5}{2 (1 + x) (1 - x)}$

Step 3.4.4.6.4

Simplify the numerator.

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Step 3.4.4.6.4.1

Add $- 1$ and $5$ .

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

Step 3.4.4.6.4.2

Factor $4$ out of $- 4 x^{2} + 4$ .

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Step 3.4.4.6.4.2.1

Factor $4$ out of $- 4 x^{2}$ .

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

Step 3.4.4.6.4.2.2

Factor $4$ out of $4$ .

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

Step 3.4.4.6.4.2.3

Factor $4$ out of $4 (- x^{2}) + 4 (1)$ .

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

Step 3.4.4.6.4.3

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

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

Step 3.4.4.6.4.4

Reorder $- x^{2}$ and $1^{2}$ .

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

Step 3.4.4.6.4.5

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 $b = x$ .

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

Step 3.4.4.6.5

Cancel the common factor of $4$ and $2$ .

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Step 3.4.4.6.5.1

Factor $2$ out of $4 (1 + x) (1 - x)$ .

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

Step 3.4.4.6.5.2

Cancel the common factor.

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

Step 3.4.4.6.5.3

Rewrite the expression.

$y = 2$

Step 3.4.4.7

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

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Step 3.4.4.7.1

Simplify the numerator.

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Step 3.4.4.7.1.1

Apply the distributive property.

$y = \frac{- 1 \cdot 1 - (4 x^{2}) \pm \sqrt{{(1 + 4 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.2

Multiply $- 1$ by $1$ .

$y = \frac{- 1 - (4 x^{2}) \pm \sqrt{{(1 + 4 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.3

Multiply $4$ by $- 1$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{{(1 + 4 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.4

Rewrite ${(1 + 4 x^{2})}^{2}$ as $(1 + 4 x^{2}) (1 + 4 x^{2})$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{(1 + 4 x^{2}) (1 + 4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.5

Expand $(1 + 4 x^{2}) (1 + 4 x^{2})$ using the FOIL Method.

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Step 3.4.4.7.1.5.1

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 (1 + 4 x^{2}) + 4 x^{2} (1 + 4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.5.2

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 \cdot 1 + 1 (4 x^{2}) + 4 x^{2} (1 + 4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.5.3

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 \cdot 1 + 1 (4 x^{2}) + 4 x^{2} \cdot 1 + 4 x^{2} (4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.6

Simplify and combine like terms.

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Step 3.4.4.7.1.6.1

Simplify each term.

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Step 3.4.4.7.1.6.1.1

Multiply $1$ by $1$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 1 (4 x^{2}) + 4 x^{2} \cdot 1 + 4 x^{2} (4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.6.1.2

Multiply $4 x^{2}$ by $1$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} \cdot 1 + 4 x^{2} (4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.6.1.3

Multiply $4$ by $1$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 4 x^{2} (4 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.6.1.4

Rewrite using the commutative property of multiplication.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 4 \cdot (4 x^{2} x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.6.1.5

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

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Step 3.4.4.7.1.6.1.5.1

Move $x^{2}$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 4 \cdot (4 (x^{2} x^{2})) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.6.1.5.2

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

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 4 \cdot (4 x^{2 + 2}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.6.1.5.3

Add $2$ and $2$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 4 \cdot (4 x^{4}) - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.6.1.6

Multiply $4$ by $4$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 4 x^{2} + 4 x^{2} + 16 x^{4} - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.6.2

Add $4 x^{2}$ and $4 x^{2}$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} - 4 \cdot (1 - x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.7

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} + (- 4 \cdot 1 - 4 (- x^{2})) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.8

Multiply $- 4$ by $1$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} + (- 4 - 4 (- x^{2})) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.9

Multiply $- 1$ by $- 4$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} + (- 4 + 4 x^{2}) \cdot (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.10

Expand $(- 4 + 4 x^{2}) (- 6 - 4 x^{2})$ using the FOIL Method.

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Step 3.4.4.7.1.10.1

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} - 4 (- 6 - 4 x^{2}) + 4 x^{2} (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.10.2

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} - 4 \cdot - 6 - 4 (- 4 x^{2}) + 4 x^{2} (- 6 - 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.10.3

Apply the distributive property.

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} - 4 \cdot - 6 - 4 (- 4 x^{2}) + 4 x^{2} \cdot - 6 + 4 x^{2} (- 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.11

Simplify and combine like terms.

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Step 3.4.4.7.1.11.1

Simplify each term.

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Step 3.4.4.7.1.11.1.1

Multiply $- 4$ by $- 6$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} + 24 - 4 (- 4 x^{2}) + 4 x^{2} \cdot - 6 + 4 x^{2} (- 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.11.1.2

Multiply $- 4$ by $- 4$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{1 + 8 x^{2} + 16 x^{4} + 24 + 16 x^{2} + 4 x^{2} \cdot - 6 + 4 x^{2} (- 4 x^{2})}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.11.1.3

Multiply $- 6$ by $4$ .

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

Step 3.4.4.7.1.11.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.4.4.7.1.11.1.5

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

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Step 3.4.4.7.1.11.1.5.1

Move $x^{2}$ .

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

Step 3.4.4.7.1.11.1.5.2

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

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

Step 3.4.4.7.1.11.1.5.3

Add $2$ and $2$ .

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

Step 3.4.4.7.1.11.1.6

Multiply $4$ by $- 4$ .

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

Step 3.4.4.7.1.11.2

Subtract $24 x^{2}$ from $16 x^{2}$ .

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

Step 3.4.4.7.1.12

Add $1$ and $24$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{8 x^{2} + 16 x^{4} + 25 - 8 x^{2} - 16 x^{4}}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.13

Subtract $8 x^{2}$ from $8 x^{2}$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{16 x^{4} + 25 + 0 - 16 x^{4}}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.14

Add $16 x^{4}$ and $0$ .

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{16 x^{4} + 25 - 16 x^{4}}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.15

Subtract $16 x^{4}$ from $16 x^{4}$ .

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

Step 3.4.4.7.1.16

Add $0$ and $25$ .

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

Step 3.4.4.7.1.17

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

$y = \frac{- 1 - 4 x^{2} \pm \sqrt{5^{2}}}{2 (1 - x^{2})}$

Step 3.4.4.7.1.18

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

$y = \frac{- 1 - 4 x^{2} \pm 5}{2 (1 - x^{2})}$

Step 3.4.4.7.2

Simplify the denominator.

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Step 3.4.4.7.2.1

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

$y = \frac{- 1 - 4 x^{2} \pm 5}{2 (1^{2} - x^{2})}$

Step 3.4.4.7.2.2

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 $b = x$ .

$y = \frac{- 1 - 4 x^{2} \pm 5}{2 (1 + x) (1 - x)}$

Step 3.4.4.7.3

Change the $\pm$ to $-$ .

$y = \frac{- 1 - 4 x^{2} - 5}{2 (1 + x) (1 - x)}$

Step 3.4.4.7.4

Simplify the numerator.

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Step 3.4.4.7.4.1

Subtract $5$ from $- 1$ .

$y = \frac{- 4 x^{2} - 6}{2 (1 + x) (1 - x)}$

Step 3.4.4.7.4.2

Factor $2$ out of $- 4 x^{2} - 6$ .

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Step 3.4.4.7.4.2.1

Factor $2$ out of $- 4 x^{2}$ .

$y = \frac{2 (- 2 x^{2}) - 6}{2 (1 + x) (1 - x)}$

Step 3.4.4.7.4.2.2

Factor $2$ out of $- 6$ .

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

Step 3.4.4.7.4.2.3

Factor $2$ out of $2 (- 2 x^{2}) + 2 (- 3)$ .

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

Step 3.4.4.7.5

Cancel the common factor of $2$ .

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Step 3.4.4.7.5.1

Cancel the common factor.

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

Step 3.4.4.7.5.2

Rewrite the expression.

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

Step 3.4.4.7.6

Factor $- 1$ out of $- 2 x^{2}$ .

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

Step 3.4.4.7.7

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 3.4.4.7.8

Factor $- 1$ out of $- (2 x^{2}) - 1 (3)$ .

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

Step 3.4.4.7.9

Rewrite $- (2 x^{2} + 3)$ as $- 1 (2 x^{2} + 3)$ .

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

Step 3.4.4.7.10

Move the negative in front of the fraction.

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

Step 3.4.4.8

The final answer is the combination of both solutions.

$y = 2$

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

$y = 2$

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

$y = 2$

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

$y = 2$

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

Step 4

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = 2, - \frac{2 x^{2} + 3}{(1 + x) (1 - x)}$

Step 5

Verify if $f^{- 1} (x) = 2, - \frac{2 x^{2} + 3}{(1 + x) (1 - x)}$ is the inverse of $f (x) = \sqrt{\frac{x + 3}{x - 2}}$ .

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Step 5.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = \sqrt{\frac{x + 3}{x - 2}}$ and $f^{- 1} (x) = 2, - \frac{2 x^{2} + 3}{(1 + x) (1 - x)}$ and compare them.

Step 5.2

Find the range of $f (x) = \sqrt{\frac{x + 3}{x - 2}}$ .

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Step 5.2.1

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[0, 1) \cup (1, \infty)$

Step 5.3

Find the domain of the inverse.

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Step 5.3.1

Find the domain of $2$ .

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Step 5.3.1.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

$(- \infty, \infty)$

Step 5.3.2

Find the domain of $- \frac{2 x^{2} + 3}{(1 + x) (1 - x)}$ .

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Step 5.3.2.1

Set the denominator in $\frac{2 x^{2} + 3}{(1 + x) (1 - x)}$ equal to $0$ to find where the expression is undefined.

$(1 + x) (1 - x) = 0$

Step 5.3.2.2

Solve for $x$ .

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Step 5.3.2.2.1

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

$1 + x = 0$

$1 - x = 0$

Step 5.3.2.2.2

Set $1 + x$ equal to $0$ and solve for $x$ .

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Step 5.3.2.2.2.1

Set $1 + x$ equal to $0$ .

$1 + x = 0$

Step 5.3.2.2.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5.3.2.2.3

Set $1 - x$ equal to $0$ and solve for $x$ .

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Step 5.3.2.2.3.1

Set $1 - x$ equal to $0$ .

$1 - x = 0$

Step 5.3.2.2.3.2

Solve $1 - x = 0$ for $x$ .

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Step 5.3.2.2.3.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 5.3.2.2.3.2.2

Divide each term in $- x = - 1$ by $- 1$ and simplify.

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Step 5.3.2.2.3.2.2.1

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Step 5.3.2.2.3.2.2.2

Simplify the left side.

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Step 5.3.2.2.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Step 5.3.2.2.3.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Step 5.3.2.2.3.2.2.3

Simplify the right side.

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Step 5.3.2.2.3.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 5.3.2.2.4

The final solution is all the values that make $(1 + x) (1 - x) = 0$ true.

$x = - 1, 1$

Step 5.3.2.3

The domain is all values of $x$ that make the expression defined.

$(- \infty, - 1) \cup (- 1, 1) \cup (1, \infty)$

Step 5.3.3

Find the union of $(- \infty, \infty), (- \infty, - 1) \cup (- 1, 1) \cup (1, \infty)$ .

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Step 5.3.3.1

The union consists of all of the elements that are contained in each interval.

$(- \infty, \infty)$

Step 5.4

Since the domain of $f^{- 1} (x) = 2, - \frac{2 x^{2} + 3}{(1 + x) (1 - x)}$ is not equal to the range of $f (x) = \sqrt{\frac{x + 3}{x - 2}}$ , then $f^{- 1} (x) = 2, - \frac{2 x^{2} + 3}{(1 + x) (1 - x)}$ is not an inverse of $f (x) = \sqrt{\frac{x + 3}{x - 2}}$ .

There is no inverse

Step 6