Solve the Rational Equation for x square root of x+ square root of x-8=2

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

Step 1

Subtract $\sqrt{x - 8}$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x^{1} = {(2 - \sqrt{x - 8})}^{2}$

Simplify.

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

Simplify the right side.

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

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

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

Expand $(2 - \sqrt{x - 8}) (2 - \sqrt{x - 8})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

$x = 2 \cdot 2 + 2 (- \sqrt{x - 8}) - \sqrt{x - 8} (2 - \sqrt{x - 8})$

Apply the distributive property.

$x = 2 \cdot 2 + 2 (- \sqrt{x - 8}) - \sqrt{x - 8} \cdot 2 - \sqrt{x - 8} (- \sqrt{x - 8})$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$x = 4 + 2 (- \sqrt{x - 8}) - \sqrt{x - 8} \cdot 2 - \sqrt{x - 8} (- \sqrt{x - 8})$

Multiply $- 1$ by $2$ .

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

Multiply $2$ by $- 1$ .

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

Multiply $- \sqrt{x - 8} (- \sqrt{x - 8})$ .

Multiply $- 1$ by $- 1$ .

$x = 4 - 2 \sqrt{x - 8} - 2 \sqrt{x - 8} + 1 \sqrt{x - 8} \sqrt{x - 8}$

Multiply $\sqrt{x - 8}$ by $1$ .

$x = 4 - 2 \sqrt{x - 8} - 2 \sqrt{x - 8} + \sqrt{x - 8} \sqrt{x - 8}$

Raise $\sqrt{x - 8}$ to the power of $1$ .

$x = 4 - 2 \sqrt{x - 8} - 2 \sqrt{x - 8} + {\sqrt{x - 8}}^{1} \sqrt{x - 8}$

Raise $\sqrt{x - 8}$ to the power of $1$ .

$x = 4 - 2 \sqrt{x - 8} - 2 \sqrt{x - 8} + {\sqrt{x - 8}}^{1} {\sqrt{x - 8}}^{1}$

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

$x = 4 - 2 \sqrt{x - 8} - 2 \sqrt{x - 8} + {\sqrt{x - 8}}^{1 + 1}$

Add $1$ and $1$ .

$x = 4 - 2 \sqrt{x - 8} - 2 \sqrt{x - 8} + {\sqrt{x - 8}}^{2}$

Rewrite ${\sqrt{x - 8}}^{2}$ as $x - 8$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 8}$ as ${(x - 8)}^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x = 4 - 2 \sqrt{x - 8} - 2 \sqrt{x - 8} + {(x - 8)}^{1}$

Simplify.

$x = 4 - 2 \sqrt{x - 8} - 2 \sqrt{x - 8} + x - 8$

Subtract $8$ from $4$ .

$x = - 4 - 2 \sqrt{x - 8} - 2 \sqrt{x - 8} + x$

Subtract $2 \sqrt{x - 8}$ from $- 2 \sqrt{x - 8}$ .

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

Step 4

Solve for $- 4 \sqrt{x - 8}$ .

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

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

Move all terms not containing $- 4 \sqrt{x - 8}$ to the right side of the equation.

Add $4$ to both sides of the equation.

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

Subtract $x$ from both sides of the equation.

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

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

Subtract $x$ from $x$ .

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

Add $0$ and $4$ .

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

Step 5

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

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

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 8}$ as ${(x - 8)}^{\frac{1}{2}}$ .

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

Simplify the left side.

Simplify ${(- 4 {(x - 8)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 4 {(x - 8)}^{\frac{1}{2}}$ .

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

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

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

Multiply the exponents in ${({(x - 8)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$16 {(x - 8)}^{1} = 4^{2}$

Simplify.

$16 (x - 8) = 4^{2}$

Apply the distributive property.

$16 x + 16 \cdot - 8 = 4^{2}$

Multiply $16$ by $- 8$ .

$16 x - 128 = 4^{2}$

Simplify the right side.

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

$16 x - 128 = 16$

Step 7

Solve for $x$ .

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

Add $128$ to both sides of the equation.

$16 x = 16 + 128$

Add $16$ and $128$ .

$16 x = 144$

Divide each term in $16 x = 144$ by $16$ and simplify.

Divide each term in $16 x = 144$ by $16$ .

$\frac{16 x}{16} = \frac{144}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 x}{16} = \frac{144}{16}$

Divide $x$ by $1$ .

$x = \frac{144}{16}$

Simplify the right side.

Divide $144$ by $16$ .

$x = 9$

Step 8

Exclude the solutions that do not make $\sqrt{x} + \sqrt{x - 8} = 2$ true.

$No solution$

Solve the Rational Equation for x square root of x+ square root of x-8=2

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