Solve for x square root of 9x-2 = square root of x+12

$\sqrt{9 x} - 2 = \sqrt{x + 12}$

Add $2$ to both sides of the equation.

$\sqrt{9 x} = \sqrt{x + 12} + 2$

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

${\sqrt{9 x}}^{2} = {(\sqrt{x + 12} + 2)}^{2}$

Simplify each side of the equation.

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

${({(9 x)}^{\frac{1}{2}})}^{2} = {(\sqrt{x + 12} + 2)}^{2}$

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, ${(a^{m})}^{n} = a^{m n}$ .

${(9 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 12} + 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(9 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 12} + 2)}^{2}$

Rewrite the expression.

${(9 x)}^{1} = {(\sqrt{x + 12} + 2)}^{2}$

Simplify.

$9 x = {(\sqrt{x + 12} + 2)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{x + 12} + 2)}^{2}$ .

Rewrite ${(\sqrt{x + 12} + 2)}^{2}$ as $(\sqrt{x + 12} + 2) (\sqrt{x + 12} + 2)$ .

$9 x = (\sqrt{x + 12} + 2) (\sqrt{x + 12} + 2)$

Expand $(\sqrt{x + 12} + 2) (\sqrt{x + 12} + 2)$ using the FOIL Method.

Apply the distributive property.

$9 x = \sqrt{x + 12} (\sqrt{x + 12} + 2) + 2 (\sqrt{x + 12} + 2)$

Apply the distributive property.

$9 x = \sqrt{x + 12} \sqrt{x + 12} + \sqrt{x + 12} \cdot 2 + 2 (\sqrt{x + 12} + 2)$

Apply the distributive property.

$9 x = \sqrt{x + 12} \sqrt{x + 12} + \sqrt{x + 12} \cdot 2 + 2 \sqrt{x + 12} + 2 \cdot 2$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x + 12} \sqrt{x + 12}$ .

Raise $\sqrt{x + 12}$ to the power of $1$ .

$9 x = {\sqrt{x + 12}}^{1} \sqrt{x + 12} + \sqrt{x + 12} \cdot 2 + 2 \sqrt{x + 12} + 2 \cdot 2$

Raise $\sqrt{x + 12}$ to the power of $1$ .

$9 x = {\sqrt{x + 12}}^{1} {\sqrt{x + 12}}^{1} + \sqrt{x + 12} \cdot 2 + 2 \sqrt{x + 12} + 2 \cdot 2$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$9 x = {\sqrt{x + 12}}^{1 + 1} + \sqrt{x + 12} \cdot 2 + 2 \sqrt{x + 12} + 2 \cdot 2$

Add $1$ and $1$ .

$9 x = {\sqrt{x + 12}}^{2} + \sqrt{x + 12} \cdot 2 + 2 \sqrt{x + 12} + 2 \cdot 2$

Rewrite ${\sqrt{x + 12}}^{2}$ as $x + 12$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 12}$ as ${(x + 12)}^{\frac{1}{2}}$ .

$9 x = {({(x + 12)}^{\frac{1}{2}})}^{2} + \sqrt{x + 12} \cdot 2 + 2 \sqrt{x + 12} + 2 \cdot 2$

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

$9 x = {(x + 12)}^{\frac{1}{2} \cdot 2} + \sqrt{x + 12} \cdot 2 + 2 \sqrt{x + 12} + 2 \cdot 2$

Combine $\frac{1}{2}$ and $2$ .

$9 x = {(x + 12)}^{\frac{2}{2}} + \sqrt{x + 12} \cdot 2 + 2 \sqrt{x + 12} + 2 \cdot 2$

Cancel the common factor of $2$ .

Cancel the common factor.

$9 x = {(x + 12)}^{\frac{2}{2}} + \sqrt{x + 12} \cdot 2 + 2 \sqrt{x + 12} + 2 \cdot 2$

Rewrite the expression.

$9 x = {(x + 12)}^{1} + \sqrt{x + 12} \cdot 2 + 2 \sqrt{x + 12} + 2 \cdot 2$

Simplify.

$9 x = x + 12 + \sqrt{x + 12} \cdot 2 + 2 \sqrt{x + 12} + 2 \cdot 2$

Move $2$ to the left of $\sqrt{x + 12}$ .

$9 x = x + 12 + 2 \cdot \sqrt{x + 12} + 2 \sqrt{x + 12} + 2 \cdot 2$

Multiply $2$ by $2$ .

$9 x = x + 12 + 2 \sqrt{x + 12} + 2 \sqrt{x + 12} + 4$

Add $12$ and $4$ .

$9 x = x + 16 + 2 \sqrt{x + 12} + 2 \sqrt{x + 12}$

Add $2 \sqrt{x + 12}$ and $2 \sqrt{x + 12}$ .

$9 x = x + 16 + 4 \sqrt{x + 12}$

Solve for $4 \sqrt{x + 12}$ .

Rewrite the equation as $x + 16 + 4 \sqrt{x + 12} = 9 x$ .

$x + 16 + 4 \sqrt{x + 12} = 9 x$

Move all terms not containing $4 \sqrt{x + 12}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$16 + 4 \sqrt{x + 12} = 9 x - x$

Subtract $16$ from both sides of the equation.

$4 \sqrt{x + 12} = 9 x - x - 16$

Subtract $x$ from $9 x$ .

$4 \sqrt{x + 12} = 8 x - 16$

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

${(4 \sqrt{x + 12})}^{2} = {(8 x - 16)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 12}$ as ${(x + 12)}^{\frac{1}{2}}$ .

${(4 {(x + 12)}^{\frac{1}{2}})}^{2} = {(8 x - 16)}^{2}$

Simplify the left side.

Simplify ${(4 {(x + 12)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $4 {(x + 12)}^{\frac{1}{2}}$ .

$4^{2} {({(x + 12)}^{\frac{1}{2}})}^{2} = {(8 x - 16)}^{2}$

Raise $4$ to the power of $2$ .

$16 {({(x + 12)}^{\frac{1}{2}})}^{2} = {(8 x - 16)}^{2}$

Multiply the exponents in ${({(x + 12)}^{\frac{1}{2}})}^{2}$ .

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

$16 {(x + 12)}^{\frac{1}{2} \cdot 2} = {(8 x - 16)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(x + 12)}^{\frac{1}{2} \cdot 2} = {(8 x - 16)}^{2}$

Rewrite the expression.

$16 {(x + 12)}^{1} = {(8 x - 16)}^{2}$

Simplify.

$16 (x + 12) = {(8 x - 16)}^{2}$

Apply the distributive property.

$16 x + 16 \cdot 12 = {(8 x - 16)}^{2}$

Multiply $16$ by $12$ .

$16 x + 192 = {(8 x - 16)}^{2}$

Simplify the right side.

Simplify ${(8 x - 16)}^{2}$ .

Rewrite ${(8 x - 16)}^{2}$ as $(8 x - 16) (8 x - 16)$ .

$16 x + 192 = (8 x - 16) (8 x - 16)$

Expand $(8 x - 16) (8 x - 16)$ using the FOIL Method.

Apply the distributive property.

$16 x + 192 = 8 x (8 x - 16) - 16 (8 x - 16)$

Apply the distributive property.

$16 x + 192 = 8 x (8 x) + 8 x \cdot - 16 - 16 (8 x - 16)$

Apply the distributive property.

$16 x + 192 = 8 x (8 x) + 8 x \cdot - 16 - 16 (8 x) - 16 \cdot - 16$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$16 x + 192 = 8 \cdot 8 x \cdot x + 8 x \cdot - 16 - 16 (8 x) - 16 \cdot - 16$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$16 x + 192 = 8 \cdot 8 (x \cdot x) + 8 x \cdot - 16 - 16 (8 x) - 16 \cdot - 16$

Multiply $x$ by $x$ .

$16 x + 192 = 8 \cdot 8 x^{2} + 8 x \cdot - 16 - 16 (8 x) - 16 \cdot - 16$

Multiply $8$ by $8$ .

$16 x + 192 = 64 x^{2} + 8 x \cdot - 16 - 16 (8 x) - 16 \cdot - 16$

Multiply $- 16$ by $8$ .

$16 x + 192 = 64 x^{2} - 128 x - 16 (8 x) - 16 \cdot - 16$

Multiply $8$ by $- 16$ .

$16 x + 192 = 64 x^{2} - 128 x - 128 x - 16 \cdot - 16$

Multiply $- 16$ by $- 16$ .

$16 x + 192 = 64 x^{2} - 128 x - 128 x + 256$

Subtract $128 x$ from $- 128 x$ .

$16 x + 192 = 64 x^{2} - 256 x + 256$

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.

$64 x^{2} - 256 x + 256 = 16 x + 192$

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

Subtract $16 x$ from both sides of the equation.

$64 x^{2} - 256 x + 256 - 16 x = 192$

Subtract $16 x$ from $- 256 x$ .

$64 x^{2} - 272 x + 256 = 192$

Subtract $192$ from both sides of the equation.

$64 x^{2} - 272 x + 256 - 192 = 0$

Subtract $192$ from $256$ .

$64 x^{2} - 272 x + 64 = 0$

Factor the left side of the equation.

Factor $16$ out of $64 x^{2} - 272 x + 64$ .

Factor $16$ out of $64 x^{2}$ .

$16 (4 x^{2}) - 272 x + 64 = 0$

Factor $16$ out of $- 272 x$ .

$16 (4 x^{2}) + 16 (- 17 x) + 64 = 0$

Factor $16$ out of $64$ .

$16 (4 x^{2}) + 16 (- 17 x) + 16 (4) = 0$

Factor $16$ out of $16 (4 x^{2}) + 16 (- 17 x)$ .

$16 (4 x^{2} - 17 x) + 16 (4) = 0$

Factor $16$ out of $16 (4 x^{2} - 17 x) + 16 (4)$ .

$16 (4 x^{2} - 17 x + 4) = 0$

Factor.

Factor by grouping.

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 $a \cdot c = 4 \cdot 4 = 16$ and whose sum is $b = - 17$ .

Factor $- 17$ out of $- 17 x$ .

$16 (4 x^{2} - 17 x + 4) = 0$

Rewrite $- 17$ as $- 1$ plus $- 16$

$16 (4 x^{2} + (- 1 - 16) x + 4) = 0$

Apply the distributive property.

$16 (4 x^{2} - 1 x - 16 x + 4) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$16 ((4 x^{2} - 1 x) - 16 x + 4) = 0$

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

$16 (x (4 x - 1) - 4 (4 x - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $4 x - 1$ .

$16 ((4 x - 1) (x - 4)) = 0$

Remove unnecessary parentheses.

$16 (4 x - 1) (x - 4) = 0$

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

$4 x - 1 = 0$

$x - 4 = 0$

Set $4 x - 1$ equal to $0$ and solve for $x$ .

Set $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Solve $4 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$4 x = 1$

Divide each term in $4 x = 1$ by $4$ and simplify.

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Divide $x$ by $1$ .

$x = \frac{1}{4}$

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 $16 (4 x - 1) (x - 4) = 0$ true.

$x = \frac{1}{4}, 4$

Exclude the solutions that do not make $\sqrt{9 x} - 2 = \sqrt{x + 12}$ true.

$x = 4$

Solve for x square root of 9x-2 = square root of x+12

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