Solve by Factoring square root of x- square root of x-32=2

$\sqrt{x} - \sqrt{x - 32} = 2$

Step 1

Subtract $2$ from both sides of the equation.

$\sqrt{x} - \sqrt{x - 32} - 2 = 0$

Step 2

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

Add $\sqrt{x - 32}$ to both sides of the equation.

$\sqrt{x} - 2 = \sqrt{x - 32}$

Add $2$ to both sides of the equation.

$\sqrt{x} = \sqrt{x - 32} + 2$

Step 3

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

${\sqrt{x}}^{2} = {(\sqrt{x - 32} + 2)}^{2}$

Step 4

Simplify each side of the equation.

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

${(x^{\frac{1}{2}})}^{2} = {(\sqrt{x - 32} + 2)}^{2}$

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

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{x - 32} + 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{x - 32} + 2)}^{2}$

Rewrite the expression.

$x^{1} = {(\sqrt{x - 32} + 2)}^{2}$

Simplify.

$x = {(\sqrt{x - 32} + 2)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{x - 32} + 2)}^{2}$ .

Rewrite ${(\sqrt{x - 32} + 2)}^{2}$ as $(\sqrt{x - 32} + 2) (\sqrt{x - 32} + 2)$ .

$x = (\sqrt{x - 32} + 2) (\sqrt{x - 32} + 2)$

Expand $(\sqrt{x - 32} + 2) (\sqrt{x - 32} + 2)$ using the FOIL Method.

Apply the distributive property.

$x = \sqrt{x - 32} (\sqrt{x - 32} + 2) + 2 (\sqrt{x - 32} + 2)$

Apply the distributive property.

$x = \sqrt{x - 32} \sqrt{x - 32} + \sqrt{x - 32} \cdot 2 + 2 (\sqrt{x - 32} + 2)$

Apply the distributive property.

$x = \sqrt{x - 32} \sqrt{x - 32} + \sqrt{x - 32} \cdot 2 + 2 \sqrt{x - 32} + 2 \cdot 2$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x - 32} \sqrt{x - 32}$ .

Raise $\sqrt{x - 32}$ to the power of $1$ .

$x = {\sqrt{x - 32}}^{1} \sqrt{x - 32} + \sqrt{x - 32} \cdot 2 + 2 \sqrt{x - 32} + 2 \cdot 2$

Raise $\sqrt{x - 32}$ to the power of $1$ .

$x = {\sqrt{x - 32}}^{1} {\sqrt{x - 32}}^{1} + \sqrt{x - 32} \cdot 2 + 2 \sqrt{x - 32} + 2 \cdot 2$

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

$x = {\sqrt{x - 32}}^{1 + 1} + \sqrt{x - 32} \cdot 2 + 2 \sqrt{x - 32} + 2 \cdot 2$

Add $1$ and $1$ .

$x = {\sqrt{x - 32}}^{2} + \sqrt{x - 32} \cdot 2 + 2 \sqrt{x - 32} + 2 \cdot 2$

Rewrite ${\sqrt{x - 32}}^{2}$ as $x - 32$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 32}$ as ${(x - 32)}^{\frac{1}{2}}$ .

$x = {({(x - 32)}^{\frac{1}{2}})}^{2} + \sqrt{x - 32} \cdot 2 + 2 \sqrt{x - 32} + 2 \cdot 2$

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

$x = {(x - 32)}^{\frac{1}{2} \cdot 2} + \sqrt{x - 32} \cdot 2 + 2 \sqrt{x - 32} + 2 \cdot 2$

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

$x = {(x - 32)}^{\frac{2}{2}} + \sqrt{x - 32} \cdot 2 + 2 \sqrt{x - 32} + 2 \cdot 2$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = {(x - 32)}^{\frac{2}{2}} + \sqrt{x - 32} \cdot 2 + 2 \sqrt{x - 32} + 2 \cdot 2$

Rewrite the expression.

$x = {(x - 32)}^{1} + \sqrt{x - 32} \cdot 2 + 2 \sqrt{x - 32} + 2 \cdot 2$

Simplify.

$x = x - 32 + \sqrt{x - 32} \cdot 2 + 2 \sqrt{x - 32} + 2 \cdot 2$

Move $2$ to the left of $\sqrt{x - 32}$ .

$x = x - 32 + 2 \cdot \sqrt{x - 32} + 2 \sqrt{x - 32} + 2 \cdot 2$

Multiply $2$ by $2$ .

$x = x - 32 + 2 \sqrt{x - 32} + 2 \sqrt{x - 32} + 4$

Add $- 32$ and $4$ .

$x = x - 28 + 2 \sqrt{x - 32} + 2 \sqrt{x - 32}$

Add $2 \sqrt{x - 32}$ and $2 \sqrt{x - 32}$ .

$x = x - 28 + 4 \sqrt{x - 32}$

Step 5

Solve for $4 \sqrt{x - 32}$ .

Rewrite the equation as $x - 28 + 4 \sqrt{x - 32} = x$ .

$x - 28 + 4 \sqrt{x - 32} = x$

Move all terms not containing $4 \sqrt{x - 32}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$- 28 + 4 \sqrt{x - 32} = x - x$

Add $28$ to both sides of the equation.

$4 \sqrt{x - 32} = x - x + 28$

Subtract $x$ from $x$ .

$4 \sqrt{x - 32} = 0 + 28$

Add $0$ and $28$ .

$4 \sqrt{x - 32} = 28$

Step 6

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

${(4 \sqrt{x - 32})}^{2} = 28^{2}$

Step 7

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 32}$ as ${(x - 32)}^{\frac{1}{2}}$ .

${(4 {(x - 32)}^{\frac{1}{2}})}^{2} = 28^{2}$

Simplify the left side.

Simplify ${(4 {(x - 32)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $4 {(x - 32)}^{\frac{1}{2}}$ .

$4^{2} {({(x - 32)}^{\frac{1}{2}})}^{2} = 28^{2}$

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

$16 {({(x - 32)}^{\frac{1}{2}})}^{2} = 28^{2}$

Multiply the exponents in ${({(x - 32)}^{\frac{1}{2}})}^{2}$ .

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

$16 {(x - 32)}^{\frac{1}{2} \cdot 2} = 28^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(x - 32)}^{\frac{1}{2} \cdot 2} = 28^{2}$

Rewrite the expression.

$16 {(x - 32)}^{1} = 28^{2}$

Simplify.

$16 (x - 32) = 28^{2}$

Apply the distributive property.

$16 x + 16 \cdot - 32 = 28^{2}$

Multiply $16$ by $- 32$ .

$16 x - 512 = 28^{2}$

Simplify the right side.

Raise $28$ to the power of $2$ .

$16 x - 512 = 784$

Step 8

Solve for $x$ .

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

Add $512$ to both sides of the equation.

$16 x = 784 + 512$

Add $784$ and $512$ .

$16 x = 1296$

Divide each term in $16 x = 1296$ by $16$ and simplify.

Divide each term in $16 x = 1296$ by $16$ .

$\frac{16 x}{16} = \frac{1296}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 x}{16} = \frac{1296}{16}$

Divide $x$ by $1$ .

$x = \frac{1296}{16}$

Simplify the right side.

Divide $1296$ by $16$ .

$x = 81$

Solve by Factoring square root of x- square root of x-32=2

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