Solve for x x=8- square root of x-8

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

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.

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

Step 2

Subtract $8$ from both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

$x - 8 = (x - 8) (x - 8)$

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

Apply the distributive property.

$x - 8 = x (x - 8) - 8 (x - 8)$

Apply the distributive property.

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

Apply the distributive property.

$x - 8 = x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 8$ to the left of $x$ .

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

Multiply $- 8$ by $- 8$ .

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

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

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

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.

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

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

Subtract $x$ from both sides of the equation.

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

Subtract $x$ from $- 16 x$ .

$x^{2} - 17 x + 64 = - 8$

Add $8$ to both sides of the equation.

$x^{2} - 17 x + 64 + 8 = 0$

Add $64$ and $8$ .

$x^{2} - 17 x + 72 = 0$

Factor $x^{2} - 17 x + 72$ 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 $72$ and whose sum is $- 17$ .

$- 9, - 8$

Write the factored form using these integers.

$(x - 9) (x - 8) = 0$

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

$x - 9 = 0$

$x - 8 = 0$

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

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

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

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

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

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

The final solution is all the values that make $(x - 9) (x - 8) = 0$ true.

$x = 9, 8$

Step 6

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

$x = 8$

Solve for x x=8- square root of x-8

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