Algebra Examples

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

Step 1

Subtract $2$ from both sides of the equation.

$\sqrt{x} + \sqrt{x - 8} - 2 = 0$

Step 2

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

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Step 2.1

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

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

Step 2.2

Add $2$ to both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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Step 4.1

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

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

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

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

Step 4.2.1.1.2.2

Rewrite the expression.

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

Step 4.2.1.2

Simplify.

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

Step 4.3

Simplify the right side.

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Step 4.3.1

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

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Step 4.3.1.1

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

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

Step 4.3.1.2

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

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Step 4.3.1.2.1

Apply the distributive property.

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

Step 4.3.1.2.2

Apply the distributive property.

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

Step 4.3.1.2.3

Apply the distributive property.

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

Step 4.3.1.3

Simplify and combine like terms.

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Step 4.3.1.3.1

Simplify each term.

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Step 4.3.1.3.1.1

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

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Step 4.3.1.3.1.1.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.1.3.1.1.2

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

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

Step 4.3.1.3.1.1.3

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

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

Step 4.3.1.3.1.1.4

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

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

Step 4.3.1.3.1.1.5

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

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

Step 4.3.1.3.1.1.6

Add $1$ and $1$ .

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

Step 4.3.1.3.1.2

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

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Step 4.3.1.3.1.2.1

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

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

Step 4.3.1.3.1.2.2

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

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

Step 4.3.1.3.1.2.3

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

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

Step 4.3.1.3.1.2.4

Cancel the common factor of $2$ .

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Step 4.3.1.3.1.2.4.1

Cancel the common factor.

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

Step 4.3.1.3.1.2.4.2

Rewrite the expression.

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

Step 4.3.1.3.1.2.5

Simplify.

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

Step 4.3.1.3.1.3

Multiply $2$ by $- 1$ .

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

Step 4.3.1.3.1.4

Multiply $- 1$ by $2$ .

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

Step 4.3.1.3.1.5

Multiply $2$ by $2$ .

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

Step 4.3.1.3.2

Add $- 8$ and $4$ .

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

Step 4.3.1.3.3

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

Subtract $x$ from both sides of the equation.

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

Step 5.2.2

Add $4$ to both sides of the equation.

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

Step 5.2.3

Subtract $x$ from $x$ .

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

Step 5.2.4

Add $0$ and $4$ .

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

Step 6

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 7

Simplify each side of the equation.

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Step 7.1

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

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

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

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

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

Step 7.2.1.2

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

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

Step 7.2.1.3

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

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Step 7.2.1.3.1

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

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

Step 7.2.1.3.2

Cancel the common factor of $2$ .

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Step 7.2.1.3.2.1

Cancel the common factor.

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

Step 7.2.1.3.2.2

Rewrite the expression.

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

Step 7.2.1.4

Simplify.

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

Step 7.2.1.5

Apply the distributive property.

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

Step 7.2.1.6

Multiply $16$ by $- 8$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

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

$16 x - 128 = 16$

Step 8

Solve for $x$ .

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Step 8.1

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

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Step 8.1.1

Add $128$ to both sides of the equation.

$16 x = 16 + 128$

Step 8.1.2

Add $16$ and $128$ .

$16 x = 144$

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $16$ .

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Step 8.2.2.1.1

Cancel the common factor.

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

Step 8.2.2.1.2

Divide $x$ by $1$ .

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

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Divide $144$ by $16$ .

$x = 9$

Step 9

Exclude the solutions that do not make $\sqrt{x} + \sqrt{x - 8} - 2 = 0$ true.

$No solution$