Solve by Factoring square root of y+8-8=y

$\sqrt{y + 8} - 8 = y$

Step 1

Subtract $y$ from both sides of the equation.

$\sqrt{y + 8} - 8 - y = 0$

Step 2

Move all terms not containing $\sqrt{y + 8}$ to the right side of the equation.

Add $8$ to both sides of the equation.

$\sqrt{y + 8} - y = 8$

Add $y$ to both sides of the equation.

$\sqrt{y + 8} = 8 + y$

Step 3

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

${\sqrt{y + 8}}^{2} = {(8 + y)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y + 8}$ as ${(y + 8)}^{\frac{1}{2}}$ .

${({(y + 8)}^{\frac{1}{2}})}^{2} = {(8 + y)}^{2}$

Simplify the left side.

Simplify ${({(y + 8)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(y + 8)}^{\frac{1}{2}})}^{2}$ .

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

${(y + 8)}^{\frac{1}{2} \cdot 2} = {(8 + y)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(y + 8)}^{\frac{1}{2} \cdot 2} = {(8 + y)}^{2}$

Rewrite the expression.

${(y + 8)}^{1} = {(8 + y)}^{2}$

Simplify.

$y + 8 = {(8 + y)}^{2}$

Simplify the right side.

Simplify ${(8 + y)}^{2}$ .

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

$y + 8 = (8 + y) (8 + y)$

Expand $(8 + y) (8 + y)$ using the FOIL Method.

Apply the distributive property.

$y + 8 = 8 (8 + y) + y (8 + y)$

Apply the distributive property.

$y + 8 = 8 \cdot 8 + 8 y + y (8 + y)$

Apply the distributive property.

$y + 8 = 8 \cdot 8 + 8 y + y \cdot 8 + y \cdot y$

Simplify and combine like terms.

Simplify each term.

Multiply $8$ by $8$ .

$y + 8 = 64 + 8 y + y \cdot 8 + y \cdot y$

Move $8$ to the left of $y$ .

$y + 8 = 64 + 8 y + 8 \cdot y + y \cdot y$

Multiply $y$ by $y$ .

$y + 8 = 64 + 8 y + 8 y + y^{2}$

Add $8 y$ and $8 y$ .

$y + 8 = 64 + 16 y + y^{2}$

Step 5

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.

$64 + 16 y + y^{2} = y + 8$

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

Subtract $y$ from both sides of the equation.

$64 + 16 y + y^{2} - y = 8$

Subtract $y$ from $16 y$ .

$64 + y^{2} + 15 y = 8$

Subtract $8$ from both sides of the equation.

$64 + y^{2} + 15 y - 8 = 0$

Subtract $8$ from $64$ .

$y^{2} + 15 y + 56 = 0$

Factor $y^{2} + 15 y + 56$ 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 $56$ and whose sum is $15$ .

$7, 8$

Write the factored form using these integers.

$(y + 7) (y + 8) = 0$

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

$y + 7 = 0$

$y + 8 = 0$

Set $y + 7$ equal to $0$ and solve for $y$ .

Set $y + 7$ equal to $0$ .

$y + 7 = 0$

Subtract $7$ from both sides of the equation.

$y = - 7$

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

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

$y + 8 = 0$

Subtract $8$ from both sides of the equation.

$y = - 8$

The final solution is all the values that make $(y + 7) (y + 8) = 0$ true.

$y = - 7, - 8$

Solve by Factoring square root of y+8-8=y

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