Solve for y y-2 = square root of 2y+20

$y - 2 = \sqrt{2 y + 20}$

Step 1

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

$\sqrt{2 y + 20} = y - 2$

Step 2

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

${\sqrt{2 y + 20}}^{2} = {(y - 2)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 y + 20}$ as ${(2 y + 20)}^{\frac{1}{2}}$ .

${({(2 y + 20)}^{\frac{1}{2}})}^{2} = {(y - 2)}^{2}$

Simplify the left side.

Simplify ${({(2 y + 20)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(2 y + 20)}^{\frac{1}{2}})}^{2}$ .

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

${(2 y + 20)}^{\frac{1}{2} \cdot 2} = {(y - 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 y + 20)}^{\frac{1}{2} \cdot 2} = {(y - 2)}^{2}$

Rewrite the expression.

${(2 y + 20)}^{1} = {(y - 2)}^{2}$

Simplify.

$2 y + 20 = {(y - 2)}^{2}$

Simplify the right side.

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

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

$2 y + 20 = (y - 2) (y - 2)$

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

Apply the distributive property.

$2 y + 20 = y (y - 2) - 2 (y - 2)$

Apply the distributive property.

$2 y + 20 = y \cdot y + y \cdot - 2 - 2 (y - 2)$

Apply the distributive property.

$2 y + 20 = y \cdot y + y \cdot - 2 - 2 y - 2 \cdot - 2$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$2 y + 20 = y^{2} + y \cdot - 2 - 2 y - 2 \cdot - 2$

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

$2 y + 20 = y^{2} - 2 \cdot y - 2 y - 2 \cdot - 2$

Multiply $- 2$ by $- 2$ .

$2 y + 20 = y^{2} - 2 y - 2 y + 4$

Subtract $2 y$ from $- 2 y$ .

$2 y + 20 = y^{2} - 4 y + 4$

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.

$y^{2} - 4 y + 4 = 2 y + 20$

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

Subtract $2 y$ from both sides of the equation.

$y^{2} - 4 y + 4 - 2 y = 20$

Subtract $2 y$ from $- 4 y$ .

$y^{2} - 6 y + 4 = 20$

Subtract $20$ from both sides of the equation.

$y^{2} - 6 y + 4 - 20 = 0$

Subtract $20$ from $4$ .

$y^{2} - 6 y - 16 = 0$

Factor $y^{2} - 6 y - 16$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 16$ and whose sum is $- 6$ .

$- 8, 2$

Write the factored form using these integers.

$(y - 8) (y + 2) = 0$

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

$y - 8 = 0$

$y + 2 = 0$

Set $y - 8$ equal to $0$ and solve for $y$ .

Set $y - 8$ equal to $0$ .

$y - 8 = 0$

Add $8$ to both sides of the equation.

$y = 8$

Set $y + 2$ equal to $0$ and solve for $y$ .

Set $y + 2$ equal to $0$ .

$y + 2 = 0$

Subtract $2$ from both sides of the equation.

$y = - 2$

The final solution is all the values that make $(y - 8) (y + 2) = 0$ true.

$y = 8, - 2$

Step 5

Exclude the solutions that do not make $y - 2 = \sqrt{2 y + 20}$ true.

$y = 8$

Solve for y y-2 = square root of 2y+20

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