Algebra Examples

$\sqrt{x - 5} - \sqrt{x} = - 2$

Step 1

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

$\sqrt{x - 5} = - 2 + \sqrt{x}$

Step 2

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

${\sqrt{x - 5}}^{2} = {(- 2 + \sqrt{x})}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 5}$ as ${(x - 5)}^{\frac{1}{2}}$ .

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 2 + \sqrt{x})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(x - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(x - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(x - 5)}^{\frac{1}{2} \cdot 2} = {(- 2 + \sqrt{x})}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(x - 5)}^{\frac{1}{2} \cdot 2} = {(- 2 + \sqrt{x})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(x - 5)}^{1} = {(- 2 + \sqrt{x})}^{2}$

Step 3.2.1.2

Simplify.

$x - 5 = {(- 2 + \sqrt{x})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

$x - 5 = (- 2 + \sqrt{x}) (- 2 + \sqrt{x})$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Apply the distributive property.

$x - 5 = - 2 (- 2 + \sqrt{x}) + \sqrt{x} (- 2 + \sqrt{x})$

Step 3.3.1.2.2

Apply the distributive property.

$x - 5 = - 2 \cdot - 2 - 2 \sqrt{x} + \sqrt{x} (- 2 + \sqrt{x})$

Step 3.3.1.2.3

Apply the distributive property.

$x - 5 = - 2 \cdot - 2 - 2 \sqrt{x} + \sqrt{x} \cdot - 2 + \sqrt{x} \sqrt{x}$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $- 2$ by $- 2$ .

$x - 5 = 4 - 2 \sqrt{x} + \sqrt{x} \cdot - 2 + \sqrt{x} \sqrt{x}$

Step 3.3.1.3.1.2

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

$x - 5 = 4 - 2 \sqrt{x} - 2 \cdot \sqrt{x} + \sqrt{x} \sqrt{x}$

Step 3.3.1.3.1.3

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

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Step 3.3.1.3.1.3.1

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

$x - 5 = 4 - 2 \sqrt{x} - 2 \sqrt{x} + {\sqrt{x}}^{1} \sqrt{x}$

Step 3.3.1.3.1.3.2

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

$x - 5 = 4 - 2 \sqrt{x} - 2 \sqrt{x} + {\sqrt{x}}^{1} {\sqrt{x}}^{1}$

Step 3.3.1.3.1.3.3

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

$x - 5 = 4 - 2 \sqrt{x} - 2 \sqrt{x} + {\sqrt{x}}^{1 + 1}$

Step 3.3.1.3.1.3.4

Add $1$ and $1$ .

$x - 5 = 4 - 2 \sqrt{x} - 2 \sqrt{x} + {\sqrt{x}}^{2}$

Step 3.3.1.3.1.4

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

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Step 3.3.1.3.1.4.1

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

$x - 5 = 4 - 2 \sqrt{x} - 2 \sqrt{x} + {(x^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.4.2

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

$x - 5 = 4 - 2 \sqrt{x} - 2 \sqrt{x} + x^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.4.3

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

$x - 5 = 4 - 2 \sqrt{x} - 2 \sqrt{x} + x^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.4.4.1

Cancel the common factor.

$x - 5 = 4 - 2 \sqrt{x} - 2 \sqrt{x} + x^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4.2

Rewrite the expression.

$x - 5 = 4 - 2 \sqrt{x} - 2 \sqrt{x} + x^{1}$

Step 3.3.1.3.1.4.5

Simplify.

$x - 5 = 4 - 2 \sqrt{x} - 2 \sqrt{x} + x$

Step 3.3.1.3.2

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

$x - 5 = 4 - 4 \sqrt{x} + x$

Step 4

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

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Step 4.1

Rewrite the equation as $4 - 4 \sqrt{x} + x = x - 5$ .

$4 - 4 \sqrt{x} + x = x - 5$

Step 4.2

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

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Step 4.2.1

Subtract $4$ from both sides of the equation.

$- 4 \sqrt{x} + x = x - 5 - 4$

Step 4.2.2

Subtract $x$ from both sides of the equation.

$- 4 \sqrt{x} = x - 5 - 4 - x$

Step 4.2.3

Combine the opposite terms in $x - 5 - 4 - x$ .

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Step 4.2.3.1

Subtract $x$ from $x$ .

$- 4 \sqrt{x} = 0 - 5 - 4$

Step 4.2.3.2

Subtract $5$ from $0$ .

$- 4 \sqrt{x} = - 5 - 4$

Step 4.2.4

Subtract $4$ from $- 5$ .

$- 4 \sqrt{x} = - 9$

Step 5

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

${(- 4 \sqrt{x})}^{2} = {(- 9)}^{2}$

Step 6

Simplify each side of the equation.

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Step 6.1

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

${(- 4 x^{\frac{1}{2}})}^{2} = {(- 9)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

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

${(- 4)}^{2} {(x^{\frac{1}{2}})}^{2} = {(- 9)}^{2}$

Step 6.2.1.2

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

$16 {(x^{\frac{1}{2}})}^{2} = {(- 9)}^{2}$

Step 6.2.1.3

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

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Step 6.2.1.3.1

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

$16 x^{\frac{1}{2} \cdot 2} = {(- 9)}^{2}$

Step 6.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.2.1.3.2.1

Cancel the common factor.

$16 x^{\frac{1}{2} \cdot 2} = {(- 9)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$16 x^{1} = {(- 9)}^{2}$

Step 6.2.1.4

Simplify.

$16 x = {(- 9)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Raise $- 9$ to the power of $2$ .

$16 x = 81$

Step 7

Divide each term in $16 x = 81$ by $16$ and simplify.

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Step 7.1

Divide each term in $16 x = 81$ by $16$ .

$\frac{16 x}{16} = \frac{81}{16}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $16$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{16 x}{16} = \frac{81}{16}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{81}{16}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{81}{16}$

Decimal Form:

$x = 5.0625$

Mixed Number Form:

$x = 5 \frac{1}{16}$