Solve for x x-10 = square root of 9x

$x - 10 = \sqrt{9 x}$

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

$\sqrt{9 x} = x - 10$

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

${\sqrt{9 x}}^{2} = {(x - 10)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 x}$ as ${(9 x)}^{\frac{1}{2}}$ .

${({(9 x)}^{\frac{1}{2}})}^{2} = {(x - 10)}^{2}$

Simplify the left side.

Simplify ${({(9 x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(9 x)}^{\frac{1}{2}})}^{2}$ .

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

${(9 x)}^{\frac{1}{2} \cdot 2} = {(x - 10)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(9 x)}^{\frac{1}{2} \cdot 2} = {(x - 10)}^{2}$

Rewrite the expression.

${(9 x)}^{1} = {(x - 10)}^{2}$

Simplify.

$9 x = {(x - 10)}^{2}$

Simplify the right side.

Simplify ${(x - 10)}^{2}$ .

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

$9 x = (x - 10) (x - 10)$

Expand $(x - 10) (x - 10)$ using the FOIL Method.

Apply the distributive property.

$9 x = x (x - 10) - 10 (x - 10)$

Apply the distributive property.

$9 x = x \cdot x + x \cdot - 10 - 10 (x - 10)$

Apply the distributive property.

$9 x = x \cdot x + x \cdot - 10 - 10 x - 10 \cdot - 10$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$9 x = x^{2} + x \cdot - 10 - 10 x - 10 \cdot - 10$

Move $- 10$ to the left of $x$ .

$9 x = x^{2} - 10 \cdot x - 10 x - 10 \cdot - 10$

Multiply $- 10$ by $- 10$ .

$9 x = x^{2} - 10 x - 10 x + 100$

Subtract $10 x$ from $- 10 x$ .

$9 x = x^{2} - 20 x + 100$

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.

$x^{2} - 20 x + 100 = 9 x$

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

Subtract $9 x$ from both sides of the equation.

$x^{2} - 20 x + 100 - 9 x = 0$

Subtract $9 x$ from $- 20 x$ .

$x^{2} - 29 x + 100 = 0$

Factor $x^{2} - 29 x + 100$ 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 $100$ and whose sum is $- 29$ .

$- 25, - 4$

Write the factored form using these integers.

$(x - 25) (x - 4) = 0$

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

$x - 25 = 0$

$x - 4 = 0$

Set $x - 25$ equal to $0$ and solve for $x$ .

Set $x - 25$ equal to $0$ .

$x - 25 = 0$

Add $25$ to both sides of the equation.

$x = 25$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make $(x - 25) (x - 4) = 0$ true.

$x = 25, 4$

Exclude the solutions that do not make $x - 10 = \sqrt{9 x}$ true.

$x = 25$

Solve for x x-10 = square root of 9x

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