Solve for x x-9 = square root of x-7

$x - 9 = \sqrt{x - 7}$

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.

$\sqrt{x - 7} = x - 9$

Step 2

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

${\sqrt{x - 7}}^{2} = {(x - 9)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 7}$ as ${(x - 7)}^{\frac{1}{2}}$ .

${({(x - 7)}^{\frac{1}{2}})}^{2} = {(x - 9)}^{2}$

Simplify the left side.

Simplify ${({(x - 7)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x - 7)}^{\frac{1}{2}})}^{2}$ .

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

${(x - 7)}^{\frac{1}{2} \cdot 2} = {(x - 9)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 7)}^{\frac{1}{2} \cdot 2} = {(x - 9)}^{2}$

Rewrite the expression.

${(x - 7)}^{1} = {(x - 9)}^{2}$

Simplify.

$x - 7 = {(x - 9)}^{2}$

Simplify the right side.

Simplify ${(x - 9)}^{2}$ .

Rewrite ${(x - 9)}^{2}$ as $(x - 9) (x - 9)$ .

$x - 7 = (x - 9) (x - 9)$

Expand $(x - 9) (x - 9)$ using the FOIL Method.

Apply the distributive property.

$x - 7 = x (x - 9) - 9 (x - 9)$

Apply the distributive property.

$x - 7 = x \cdot x + x \cdot - 9 - 9 (x - 9)$

Apply the distributive property.

$x - 7 = x \cdot x + x \cdot - 9 - 9 x - 9 \cdot - 9$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x - 7 = x^{2} + x \cdot - 9 - 9 x - 9 \cdot - 9$

Move $- 9$ to the left of $x$ .

$x - 7 = x^{2} - 9 \cdot x - 9 x - 9 \cdot - 9$

Multiply $- 9$ by $- 9$ .

$x - 7 = x^{2} - 9 x - 9 x + 81$

Subtract $9 x$ from $- 9 x$ .

$x - 7 = x^{2} - 18 x + 81$

Step 4

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.

$x^{2} - 18 x + 81 = x - 7$

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

Subtract $x$ from both sides of the equation.

$x^{2} - 18 x + 81 - x = - 7$

Subtract $x$ from $- 18 x$ .

$x^{2} - 19 x + 81 = - 7$

Add $7$ to both sides of the equation.

$x^{2} - 19 x + 81 + 7 = 0$

Add $81$ and $7$ .

$x^{2} - 19 x + 88 = 0$

Factor $x^{2} - 19 x + 88$ 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 $88$ and whose sum is $- 19$ .

$- 11, - 8$

Write the factored form using these integers.

$(x - 11) (x - 8) = 0$

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

$x - 11 = 0$

$x - 8 = 0$

Set $x - 11$ equal to $0$ and solve for $x$ .

Set $x - 11$ equal to $0$ .

$x - 11 = 0$

Add $11$ to both sides of the equation.

$x = 11$

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 $(x - 11) (x - 8) = 0$ true.

$x = 11, 8$

Step 5

Exclude the solutions that do not make $x - 9 = \sqrt{x - 7}$ true.

$x = 11$

Solve for x x-9 = square root of x-7

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