Solve for y x = square root of (y+1)/(y-2)

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

Step 1

Multiply the equation by $y - 2$ .

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

Step 2

Simplify the left side.

Apply the distributive property.

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

Step 3

Simplify the right side.

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

Rewrite $\sqrt{\frac{y + 1}{y - 2}}$ as $\frac{\sqrt{y + 1}}{\sqrt{y - 2}}$ .

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

Multiply $\frac{\sqrt{y + 1}}{\sqrt{y - 2}}$ by $\frac{\sqrt{y - 2}}{\sqrt{y - 2}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{\sqrt{y + 1}}{\sqrt{y - 2}}$ by $\frac{\sqrt{y - 2}}{\sqrt{y - 2}}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Combine using the product rule for radicals.

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

Cancel the common factor of $y - 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Solve for $y$ .

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

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Expand $(y + 1) (y - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

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

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

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

Multiply $y$ by $1$ .

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

Multiply $- 2$ by $1$ .

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

Add $- 2 y$ and $y$ .

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $y$ .

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

Multiply $y$ by $y$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 2$ .

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

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

Move $y$ .

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

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

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

Solve for $y$ .

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

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

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

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

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

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

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 1$ by $- 1$ .

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

Multiply $4$ by $- 1$ .

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $- 1$ by $- 1$ .

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

Multiply $4$ by $- 1$ .

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

Multiply $- 1$ by $4$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Multiply $4$ by $4$ .

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

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

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

Apply the distributive property.

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

Multiply $- 4$ by $1$ .

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

Multiply $- 1$ by $- 4$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $- 4$ by $- 2$ .

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

Multiply $- 4$ by $- 4$ .

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

Multiply $- 2$ by $4$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Multiply $4$ by $- 4$ .

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

Subtract $8 x^{2}$ from $16 x^{2}$ .

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

Add $1$ and $8$ .

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

Add $- 8 x^{2}$ and $8 x^{2}$ .

$y = \frac{1 - 4 x^{2} \pm \sqrt{16 x^{4} + 9 + 0 - 16 x^{4}}}{2 (1 - x^{2})}$

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

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

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

$y = \frac{1 - 4 x^{2} \pm \sqrt{0 + 9}}{2 (1 - x^{2})}$

Add $0$ and $9$ .

$y = \frac{1 - 4 x^{2} \pm \sqrt{9}}{2 (1 - x^{2})}$

Rewrite $9$ as $3^{2}$ .

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

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

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

Simplify the denominator.

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

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

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 $b = x$ .

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

The final answer is the combination of both solutions.

$y = 2$

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

$y = 2$

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

$y = 2$

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

Solve for y x = square root of (y+1)/(y-2)

Related Questions

Support LCMGCF.com by disabling your adblocker.