Solve by Factoring square root of x+ square root of x-12=2

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

Step 1

Subtract $2$ from both sides of the equation.

$\sqrt{x} + \sqrt{x - 12} - 2 = 0$

Step 2

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

Subtract $\sqrt{x - 12}$ from both sides of the equation.

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

Add $2$ to both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $- \sqrt{x - 12} (- \sqrt{x - 12})$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $\sqrt{x - 12}$ by $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $2$ by $2$ .

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

Add $- 12$ and $4$ .

$x = x - 8 - 2 \sqrt{x - 12} - 2 \sqrt{x - 12}$

Subtract $2 \sqrt{x - 12}$ from $- 2 \sqrt{x - 12}$ .

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

Step 5

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

Rewrite the equation as $x - 8 - 4 \sqrt{x - 12} = x$ .

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

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

Subtract $x$ from both sides of the equation.

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

Add $8$ to both sides of the equation.

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

Subtract $x$ from $x$ .

$- 4 \sqrt{x - 12} = 0 + 8$

Add $0$ and $8$ .

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

Step 6

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

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

Step 7

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply $16$ by $- 12$ .

$16 x - 192 = 8^{2}$

Simplify the right side.

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

$16 x - 192 = 64$

Step 8

Solve for $x$ .

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

Add $192$ to both sides of the equation.

$16 x = 64 + 192$

Add $64$ and $192$ .

$16 x = 256$

Divide each term in $16 x = 256$ by $16$ and simplify.

Divide each term in $16 x = 256$ by $16$ .

$\frac{16 x}{16} = \frac{256}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 x}{16} = \frac{256}{16}$

Divide $x$ by $1$ .

$x = \frac{256}{16}$

Simplify the right side.

Divide $256$ by $16$ .

$x = 16$

Step 9

Exclude the solutions that do not make $\sqrt{x} + \sqrt{x - 12} - 2 = 0$ true.

$No solution$

Solve by Factoring square root of x+ square root of x-12=2

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