Solve for x square root of x+6-2 = square root of x-2

$\sqrt{x + 6} - 2 = \sqrt{x - 2}$

Step 1

Add $2$ to both sides of the equation.

$\sqrt{x + 6} = \sqrt{x - 2} + 2$

Step 2

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

${\sqrt{x + 6}}^{2} = {(\sqrt{x - 2} + 2)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 6}$ as ${(x + 6)}^{\frac{1}{2}}$ .

${({(x + 6)}^{\frac{1}{2}})}^{2} = {(\sqrt{x - 2} + 2)}^{2}$

Simplify the left side.

Simplify ${({(x + 6)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x + 6)}^{\frac{1}{2}})}^{2}$ .

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

${(x + 6)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x - 2} + 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 6)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x - 2} + 2)}^{2}$

Rewrite the expression.

${(x + 6)}^{1} = {(\sqrt{x - 2} + 2)}^{2}$

Simplify.

$x + 6 = {(\sqrt{x - 2} + 2)}^{2}$

Simplify the right side.

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

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

$x + 6 = (\sqrt{x - 2} + 2) (\sqrt{x - 2} + 2)$

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

Apply the distributive property.

$x + 6 = \sqrt{x - 2} (\sqrt{x - 2} + 2) + 2 (\sqrt{x - 2} + 2)$

Apply the distributive property.

$x + 6 = \sqrt{x - 2} \sqrt{x - 2} + \sqrt{x - 2} \cdot 2 + 2 (\sqrt{x - 2} + 2)$

Apply the distributive property.

$x + 6 = \sqrt{x - 2} \sqrt{x - 2} + \sqrt{x - 2} \cdot 2 + 2 \sqrt{x - 2} + 2 \cdot 2$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x - 2} \sqrt{x - 2}$ .

Raise $\sqrt{x - 2}$ to the power of $1$ .

$x + 6 = {\sqrt{x - 2}}^{1} \sqrt{x - 2} + \sqrt{x - 2} \cdot 2 + 2 \sqrt{x - 2} + 2 \cdot 2$

Raise $\sqrt{x - 2}$ to the power of $1$ .

$x + 6 = {\sqrt{x - 2}}^{1} {\sqrt{x - 2}}^{1} + \sqrt{x - 2} \cdot 2 + 2 \sqrt{x - 2} + 2 \cdot 2$

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

$x + 6 = {\sqrt{x - 2}}^{1 + 1} + \sqrt{x - 2} \cdot 2 + 2 \sqrt{x - 2} + 2 \cdot 2$

Add $1$ and $1$ .

$x + 6 = {\sqrt{x - 2}}^{2} + \sqrt{x - 2} \cdot 2 + 2 \sqrt{x - 2} + 2 \cdot 2$

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

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

$x + 6 = {({(x - 2)}^{\frac{1}{2}})}^{2} + \sqrt{x - 2} \cdot 2 + 2 \sqrt{x - 2} + 2 \cdot 2$

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

$x + 6 = {(x - 2)}^{\frac{1}{2} \cdot 2} + \sqrt{x - 2} \cdot 2 + 2 \sqrt{x - 2} + 2 \cdot 2$

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

$x + 6 = {(x - 2)}^{\frac{2}{2}} + \sqrt{x - 2} \cdot 2 + 2 \sqrt{x - 2} + 2 \cdot 2$

Cancel the common factor of $2$ .

Cancel the common factor.

$x + 6 = {(x - 2)}^{\frac{2}{2}} + \sqrt{x - 2} \cdot 2 + 2 \sqrt{x - 2} + 2 \cdot 2$

Rewrite the expression.

$x + 6 = {(x - 2)}^{1} + \sqrt{x - 2} \cdot 2 + 2 \sqrt{x - 2} + 2 \cdot 2$

Simplify.

$x + 6 = x - 2 + \sqrt{x - 2} \cdot 2 + 2 \sqrt{x - 2} + 2 \cdot 2$

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

$x + 6 = x - 2 + 2 \cdot \sqrt{x - 2} + 2 \sqrt{x - 2} + 2 \cdot 2$

Multiply $2$ by $2$ .

$x + 6 = x - 2 + 2 \sqrt{x - 2} + 2 \sqrt{x - 2} + 4$

Add $- 2$ and $4$ .

$x + 6 = x + 2 + 2 \sqrt{x - 2} + 2 \sqrt{x - 2}$

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

$x + 6 = x + 2 + 4 \sqrt{x - 2}$

Step 4

Solve for $4 \sqrt{x - 2}$ .

Rewrite the equation as $x + 2 + 4 \sqrt{x - 2} = x + 6$ .

$x + 2 + 4 \sqrt{x - 2} = x + 6$

Move all terms not containing $4 \sqrt{x - 2}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$2 + 4 \sqrt{x - 2} = x + 6 - x$

Subtract $2$ from both sides of the equation.

$4 \sqrt{x - 2} = x + 6 - x - 2$

Combine the opposite terms in $x + 6 - x - 2$ .

Subtract $x$ from $x$ .

$4 \sqrt{x - 2} = 0 + 6 - 2$

Add $0$ and $6$ .

$4 \sqrt{x - 2} = 6 - 2$

Subtract $2$ from $6$ .

$4 \sqrt{x - 2} = 4$

Step 5

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

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

Step 6

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

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

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

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

$16 {(x - 2)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(x - 2)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

$16 x + 16 \cdot - 2 = 4^{2}$

Multiply $16$ by $- 2$ .

$16 x - 32 = 4^{2}$

Simplify the right side.

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

$16 x - 32 = 16$

Step 7

Solve for $x$ .

Move all terms not containing $x$ to the right side of the equation.

Add $32$ to both sides of the equation.

$16 x = 16 + 32$

Add $16$ and $32$ .

$16 x = 48$

Divide each term in $16 x = 48$ by $16$ and simplify.

Divide each term in $16 x = 48$ by $16$ .

$\frac{16 x}{16} = \frac{48}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 x}{16} = \frac{48}{16}$

Divide $x$ by $1$ .

$x = \frac{48}{16}$

Simplify the right side.

Divide $48$ by $16$ .

$x = 3$

Solve for x square root of x+6-2 = square root of x-2

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