Solve by Factoring square root of 9x=x+2

$\sqrt{9 x} = x + 2$

Step 1

Move all the expressions to the left side of the equation.

Subtract $x$ from both sides of the equation.

$\sqrt{9 x} - x = 2$

Subtract $2$ from both sides of the equation.

$\sqrt{9 x} - x - 2 = 0$

Step 2

Simplify each term.

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

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

Pull terms out from under the radical.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Factor by grouping.

Reorder terms.

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

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 2 = 2$ and whose sum is $b = 3$ .

Factor $3$ out of $3 u$ .

$- u^{2} + 3 (u) - 2 = 0$

Rewrite $3$ as $1$ plus $2$

$- u^{2} + (1 + 2) u - 2 = 0$

Apply the distributive property.

$- u^{2} + 1 u + 2 u - 2 = 0$

Multiply $u$ by $1$ .

$- u^{2} + u + 2 u - 2 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(- u^{2} + u) + 2 u - 2 = 0$

Factor out the greatest common factor (GCF) from each group.

$u (- u + 1) - 2 (- u + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $- u + 1$ .

$(- u + 1) (u - 2) = 0$

Step 7

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

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

Step 8

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

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

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

Step 9

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

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

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

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

Subtract $1$ from both sides of the equation.

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

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

$x = 1$

Step 10

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 $x$ .

Add $2$ to 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 11

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

$x = 1, 4$

Solve by Factoring square root of 9x=x+2

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