Solve by Factoring y- square root of y-6=8

$y - \sqrt{y - 6} = 8$

Step 1

Subtract $8$ from both sides of the equation.

$y - \sqrt{y - 6} - 8 = 0$

Step 2

Move all terms not containing $- \sqrt{y - 6}$ to the right side of the equation.

Subtract $y$ from both sides of the equation.

$- \sqrt{y - 6} - 8 = - y$

Add $8$ to both sides of the equation.

$- \sqrt{y - 6} = - y + 8$

Step 3

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

${(- \sqrt{y - 6})}^{2} = {(- y + 8)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y - 6}$ as ${(y - 6)}^{\frac{1}{2}}$ .

${(- {(y - 6)}^{\frac{1}{2}})}^{2} = {(- y + 8)}^{2}$

Simplify the left side.

Simplify ${(- {(y - 6)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- {(y - 6)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(y - 6)}^{\frac{1}{2}})}^{2} = {(- y + 8)}^{2}$

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

$1 {({(y - 6)}^{\frac{1}{2}})}^{2} = {(- y + 8)}^{2}$

Multiply ${({(y - 6)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(y - 6)}^{\frac{1}{2}})}^{2} = {(- y + 8)}^{2}$

Multiply the exponents in ${({(y - 6)}^{\frac{1}{2}})}^{2}$ .

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

${(y - 6)}^{\frac{1}{2} \cdot 2} = {(- y + 8)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(y - 6)}^{\frac{1}{2} \cdot 2} = {(- y + 8)}^{2}$

Rewrite the expression.

${(y - 6)}^{1} = {(- y + 8)}^{2}$

Simplify.

$y - 6 = {(- y + 8)}^{2}$

Simplify the right side.

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

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

$y - 6 = (- y + 8) (- y + 8)$

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

Apply the distributive property.

$y - 6 = - y (- y + 8) + 8 (- y + 8)$

Apply the distributive property.

$y - 6 = - y (- y) - y \cdot 8 + 8 (- y + 8)$

Apply the distributive property.

$y - 6 = - y (- y) - y \cdot 8 + 8 (- y) + 8 \cdot 8$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$y - 6 = - 1 \cdot - 1 y \cdot y - y \cdot 8 + 8 (- y) + 8 \cdot 8$

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

Move $y$ .

$y - 6 = - 1 \cdot - 1 (y \cdot y) - y \cdot 8 + 8 (- y) + 8 \cdot 8$

Multiply $y$ by $y$ .

$y - 6 = - 1 \cdot - 1 y^{2} - y \cdot 8 + 8 (- y) + 8 \cdot 8$

Multiply $- 1$ by $- 1$ .

$y - 6 = 1 y^{2} - y \cdot 8 + 8 (- y) + 8 \cdot 8$

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

$y - 6 = y^{2} - y \cdot 8 + 8 (- y) + 8 \cdot 8$

Multiply $8$ by $- 1$ .

$y - 6 = y^{2} - 8 y + 8 (- y) + 8 \cdot 8$

Multiply $- 1$ by $8$ .

$y - 6 = y^{2} - 8 y - 8 y + 8 \cdot 8$

Multiply $8$ by $8$ .

$y - 6 = y^{2} - 8 y - 8 y + 64$

Subtract $8 y$ from $- 8 y$ .

$y - 6 = y^{2} - 16 y + 64$

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.

$y^{2} - 16 y + 64 = y - 6$

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

Subtract $y$ from both sides of the equation.

$y^{2} - 16 y + 64 - y = - 6$

Subtract $y$ from $- 16 y$ .

$y^{2} - 17 y + 64 = - 6$

Add $6$ to both sides of the equation.

$y^{2} - 17 y + 64 + 6 = 0$

Add $64$ and $6$ .

$y^{2} - 17 y + 70 = 0$

Factor $y^{2} - 17 y + 70$ 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 $70$ and whose sum is $- 17$ .

$- 10, - 7$

Write the factored form using these integers.

$(y - 10) (y - 7) = 0$

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

$y - 10 = 0$

$y - 7 = 0$

Set $y - 10$ equal to $0$ and solve for $y$ .

Set $y - 10$ equal to $0$ .

$y - 10 = 0$

Add $10$ to both sides of the equation.

$y = 10$

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

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

$y - 7 = 0$

Add $7$ to both sides of the equation.

$y = 7$

The final solution is all the values that make $(y - 10) (y - 7) = 0$ true.

$y = 10, 7$

Step 6

Exclude the solutions that do not make $y - \sqrt{y - 6} - 8 = 0$ true.

$y = 10$

Solve by Factoring y- square root of y-6=8

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