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

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

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

Multiply $2$ by $2$ .

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

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

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

Solve for $4 \sqrt{x}$ .

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

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

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

Subtract $x$ from both sides of the equation.

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

Subtract $4$ from both sides of the equation.

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

Combine the opposite terms in $x + 12 - x - 4$ .

Subtract $x$ from $x$ .

$4 \sqrt{x} = 0 + 12 - 4$

Add $0$ and $12$ .

$4 \sqrt{x} = 12 - 4$

Subtract $4$ from $12$ .

$4 \sqrt{x} = 8$

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

${(4 \sqrt{x})}^{2} = 8^{2}$

Simplify each side of the equation.

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

${(4 x^{\frac{1}{2}})}^{2} = 8^{2}$

Simplify the left side.

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

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

$4^{2} {(x^{\frac{1}{2}})}^{2} = 8^{2}$

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

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

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

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

$16 x^{\frac{1}{2} \cdot 2} = 8^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 x^{\frac{1}{2} \cdot 2} = 8^{2}$

Rewrite the expression.

$16 x^{1} = 8^{2}$

Simplify.

$16 x = 8^{2}$

Simplify the right side.

Raise $8$ to the power of $2$ .

$16 x = 64$

Divide each term in $16 x = 64$ by $16$ and simplify.

Divide each term in $16 x = 64$ by $16$ .

$\frac{16 x}{16} = \frac{64}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 x}{16} = \frac{64}{16}$

Divide $x$ by $1$ .

$x = \frac{64}{16}$

Simplify the right side.

Divide $64$ by $16$ .

$x = 4$

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

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