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

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

Step 1

Subtract $\sqrt{x - 6}$ from both sides of the equation.

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

Step 2

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

Subtract $x$ from both sides of the equation.

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

Add $8$ to both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $- 1$ .

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

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

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

Multiply $8$ by $- 1$ .

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

Multiply $- 1$ by $8$ .

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

Multiply $8$ by $8$ .

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

Subtract $8 x$ from $- 8 x$ .

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

Step 5

Solve for $x$ .

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

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

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

Subtract $x$ from both sides of the equation.

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

Subtract $x$ from $- 16 x$ .

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

Add $6$ to both sides of the equation.

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

Add $64$ and $6$ .

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

Factor $x^{2} - 17 x + 70$ 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.

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

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

$x - 10 = 0$

$x - 7 = 0$

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

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

$x - 10 = 0$

Add $10$ to both sides of the equation.

$x = 10$

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

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

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

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

$x = 10, 7$

Step 6

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

$x = 10$

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

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