Algebra Examples

$\sqrt{x + 28} - 2 \sqrt{x + 1} = 0$

Step 1

Add $2 \sqrt{x + 1}$ to both sides of the equation.

$\sqrt{x + 28} = 2 \sqrt{x + 1}$

Step 2

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

${\sqrt{x + 28}}^{2} = {(2 \sqrt{x + 1})}^{2}$

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 $\sqrt{x + 28}$ as ${(x + 28)}^{\frac{1}{2}}$ .

${({(x + 28)}^{\frac{1}{2}})}^{2} = {(2 \sqrt{x + 1})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(x + 28)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(x + 28)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(x + 28)}^{\frac{1}{2} \cdot 2} = {(2 \sqrt{x + 1})}^{2}$

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.

${(x + 28)}^{\frac{1}{2} \cdot 2} = {(2 \sqrt{x + 1})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(x + 28)}^{1} = {(2 \sqrt{x + 1})}^{2}$

Step 3.2.1.2

Simplify.

$x + 28 = {(2 \sqrt{x + 1})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Simplify the expression.

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Step 3.3.1.1.1

Apply the product rule to $2 \sqrt{x + 1}$ .

$x + 28 = 2^{2} {\sqrt{x + 1}}^{2}$

Step 3.3.1.1.2

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

$x + 28 = 4 {\sqrt{x + 1}}^{2}$

Step 3.3.1.2

Rewrite ${\sqrt{x + 1}}^{2}$ as $x + 1$ .

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Step 3.3.1.2.1

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

$x + 28 = 4 {({(x + 1)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.2.2

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

$x + 28 = 4 {(x + 1)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.2.3

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

$x + 28 = 4 {(x + 1)}^{\frac{2}{2}}$

Step 3.3.1.2.4

Cancel the common factor of $2$ .

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Step 3.3.1.2.4.1

Cancel the common factor.

$x + 28 = 4 {(x + 1)}^{\frac{2}{2}}$

Step 3.3.1.2.4.2

Rewrite the expression.

$x + 28 = 4 {(x + 1)}^{1}$

Step 3.3.1.2.5

Simplify.

$x + 28 = 4 (x + 1)$

Step 3.3.1.3

Apply the distributive property.

$x + 28 = 4 x + 4 \cdot 1$

Step 3.3.1.4

Multiply $4$ by $1$ .

$x + 28 = 4 x + 4$

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 $4 x$ from both sides of the equation.

$x + 28 - 4 x = 4$

Step 4.1.2

Subtract $4 x$ from $x$ .

$- 3 x + 28 = 4$

Step 4.2

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

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Step 4.2.1

Subtract $28$ from both sides of the equation.

$- 3 x = 4 - 28$

Step 4.2.2

Subtract $28$ from $4$ .

$- 3 x = - 24$

Step 4.3

Divide each term in $- 3 x = - 24$ by $- 3$ and simplify.

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Step 4.3.1

Divide each term in $- 3 x = - 24$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 24}{- 3}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $- 3$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 24}{- 3}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 24}{- 3}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $- 24$ by $- 3$ .

$x = 8$