Solve by Factoring x-10 = square root of 9x

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

Step 1

Subtract $\sqrt{9 x}$ from both sides of the equation.

$x - 10 - \sqrt{9 x} = 0$

Step 2

Simplify each term.

Rewrite $9$ as $3^{2}$ .

$x - 10 - \sqrt{3^{2} x} = 0$

Pull terms out from under the radical.

$x - 10 - (3 \sqrt{x}) = 0$

Multiply $3$ by $- 1$ .

$x - 10 - 3 \sqrt{x} = 0$

Step 3

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

$x - 10 - 3 x^{\frac{1}{2}} = 0$

Step 4

Reorder terms.

$x - 3 x^{\frac{1}{2}} - 10 = 0$

Step 5

Rewrite $x - 3 x^{\frac{1}{2}} - 10$ in a factored form.

Rewrite $x$ as ${(x^{\frac{1}{2}})}^{2}$ .

${(x^{\frac{1}{2}})}^{2} - 3 x^{\frac{1}{2}} - 10 = 0$

Let $u = x^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{2}}$ .

$u^{2} - 3 u - 10 = 0$

Factor $u^{2} - 3 u - 10$ 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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Write the factored form using these integers.

$(u - 5) (u + 2) = 0$

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

$(x^{\frac{1}{2}} - 5) (x^{\frac{1}{2}} + 2) = 0$

Step 6

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

$x^{\frac{1}{2}} - 5 = 0$

$x^{\frac{1}{2}} + 2 = 0$

Step 7

Set $x^{\frac{1}{2}} - 5$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{2}} - 5$ equal to $0$ .

$x^{\frac{1}{2}} - 5 = 0$

Solve $x^{\frac{1}{2}} - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$x^{\frac{1}{2}} = 5$

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{2}})}^{2} = 5^{2}$

Simplify the exponent.

Simplify the left side.

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

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

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

$x^{\frac{1}{2} \cdot 2} = 5^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 5^{2}$

Rewrite the expression.

$x^{1} = 5^{2}$

Simplify.

$x = 5^{2}$

Simplify the right side.

Raise $5$ to the power of $2$ .

$x = 25$

Step 8

Set $x^{\frac{1}{2}} + 2$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{2}} + 2$ equal to $0$ .

$x^{\frac{1}{2}} + 2 = 0$

Solve $x^{\frac{1}{2}} + 2 = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

$x^{\frac{1}{2}} = - 2$

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

$x = 4$

Step 9

The final solution is all the values that make $(x^{\frac{1}{2}} - 5) (x^{\frac{1}{2}} + 2) = 0$ true.

$x = 25, 4$

Step 10

Exclude the solutions that do not make $(x^{\frac{1}{2}} - 5) (x^{\frac{1}{2}} + 2) = 0$ true.

$x = 25$

Solve by Factoring x-10 = square root of 9x

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