Solve by Factoring y- square root of y-8=10

$y - \sqrt{y - 8} = 10$

Step 1

Subtract $10$ from both sides of the equation.

$y - \sqrt{y - 8} - 10 = 0$

Step 2

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

Subtract $y$ from both sides of the equation.

$- \sqrt{y - 8} - 10 = - y$

Add $10$ to both sides of the equation.

$- \sqrt{y - 8} = - y + 10$

Step 3

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

${(- \sqrt{y - 8})}^{2} = {(- y + 10)}^{2}$

Step 4

Simplify each side of the equation.

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

${(- {(y - 8)}^{\frac{1}{2}})}^{2} = {(- y + 10)}^{2}$

Simplify the left side.

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

Apply the product rule to $- {(y - 8)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(y - 8)}^{\frac{1}{2}})}^{2} = {(- y + 10)}^{2}$

Raise $- 1$ to the power of $2$ .

$1 {({(y - 8)}^{\frac{1}{2}})}^{2} = {(- y + 10)}^{2}$

Multiply ${({(y - 8)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(y - 8)}^{\frac{1}{2}})}^{2} = {(- y + 10)}^{2}$

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

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

${(y - 8)}^{\frac{1}{2} \cdot 2} = {(- y + 10)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(y - 8)}^{\frac{1}{2} \cdot 2} = {(- y + 10)}^{2}$

Rewrite the expression.

${(y - 8)}^{1} = {(- y + 10)}^{2}$

Simplify.

$y - 8 = {(- y + 10)}^{2}$

Simplify the right side.

Simplify ${(- y + 10)}^{2}$ .

Rewrite ${(- y + 10)}^{2}$ as $(- y + 10) (- y + 10)$ .

$y - 8 = (- y + 10) (- y + 10)$

Expand $(- y + 10) (- y + 10)$ using the FOIL Method.

Apply the distributive property.

$y - 8 = - y (- y + 10) + 10 (- y + 10)$

Apply the distributive property.

$y - 8 = - y (- y) - y \cdot 10 + 10 (- y + 10)$

Apply the distributive property.

$y - 8 = - y (- y) - y \cdot 10 + 10 (- y) + 10 \cdot 10$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$y - 8 = - 1 \cdot - 1 y \cdot y - y \cdot 10 + 10 (- y) + 10 \cdot 10$

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

Move $y$ .

$y - 8 = - 1 \cdot - 1 (y \cdot y) - y \cdot 10 + 10 (- y) + 10 \cdot 10$

Multiply $y$ by $y$ .

$y - 8 = - 1 \cdot - 1 y^{2} - y \cdot 10 + 10 (- y) + 10 \cdot 10$

Multiply $- 1$ by $- 1$ .

$y - 8 = 1 y^{2} - y \cdot 10 + 10 (- y) + 10 \cdot 10$

Multiply $y^{2}$ by $1$ .

$y - 8 = y^{2} - y \cdot 10 + 10 (- y) + 10 \cdot 10$

Multiply $10$ by $- 1$ .

$y - 8 = y^{2} - 10 y + 10 (- y) + 10 \cdot 10$

Multiply $- 1$ by $10$ .

$y - 8 = y^{2} - 10 y - 10 y + 10 \cdot 10$

Multiply $10$ by $10$ .

$y - 8 = y^{2} - 10 y - 10 y + 100$

Subtract $10 y$ from $- 10 y$ .

$y - 8 = y^{2} - 20 y + 100$

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.

$y^{2} - 20 y + 100 = y - 8$

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

Subtract $y$ from both sides of the equation.

$y^{2} - 20 y + 100 - y = - 8$

Subtract $y$ from $- 20 y$ .

$y^{2} - 21 y + 100 = - 8$

Add $8$ to both sides of the equation.

$y^{2} - 21 y + 100 + 8 = 0$

Add $100$ and $8$ .

$y^{2} - 21 y + 108 = 0$

Factor $y^{2} - 21 y + 108$ 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 $108$ and whose sum is $- 21$ .

$- 12, - 9$

Write the factored form using these integers.

$(y - 12) (y - 9) = 0$

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

$y - 12 = 0$

$y - 9 = 0$

Set $y - 12$ equal to $0$ and solve for $y$ .

Set $y - 12$ equal to $0$ .

$y - 12 = 0$

Add $12$ to both sides of the equation.

$y = 12$

Set $y - 9$ equal to $0$ and solve for $y$ .

Set $y - 9$ equal to $0$ .

$y - 9 = 0$

Add $9$ to both sides of the equation.

$y = 9$

The final solution is all the values that make $(y - 12) (y - 9) = 0$ true.

$y = 12, 9$

Step 6

Exclude the solutions that do not make $y - \sqrt{y - 8} - 10 = 0$ true.

$y = 12$

Solve by Factoring y- square root of y-8=10

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