Algebra Examples

$\sqrt{2 x - 1} - \sqrt{x + 4} = 0$

Step 1

Add $\sqrt{x + 4}$ to both sides of the equation.

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

Step 2

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

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

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 $\sqrt{2 x - 1}$ as ${(2 x - 1)}^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Rewrite ${\sqrt{x + 4}}^{2}$ as $x + 4$ .

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Step 3.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 4}$ as ${(x + 4)}^{\frac{1}{2}}$ .

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

Step 3.3.1.2

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

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

Step 3.3.1.3

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

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

Step 3.3.1.4

Cancel the common factor of $2$ .

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Step 3.3.1.4.1

Cancel the common factor.

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

Step 3.3.1.4.2

Rewrite the expression.

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

Step 3.3.1.5

Simplify.

$2 x - 1 = x + 4$

Step 4

Solve for $x$ .

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Step 4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.1.1

Subtract $x$ from both sides of the equation.

$2 x - 1 - x = 4$

Step 4.1.2

Subtract $x$ from $2 x$ .

$x - 1 = 4$

Step 4.2

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

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Step 4.2.1

Add $1$ to both sides of the equation.

$x = 4 + 1$

Step 4.2.2

Add $4$ and $1$ .

$x = 5$