Algebra Examples

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

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 2

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

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

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.1.3

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

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Step 3.2.1.3.1

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

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

Step 3.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

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

Step 3.2.1.3.2.2

Rewrite the expression.

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

Step 3.2.1.4

Simplify.

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

Step 3.2.1.5

Apply the distributive property.

$16 x + 16 \cdot - 2 = {(x + 2)}^{2}$

Step 3.2.1.6

Multiply $16$ by $- 2$ .

$16 x - 32 = {(x + 2)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(x + 2)}^{2}$ .

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Step 3.3.1.1

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

$16 x - 32 = (x + 2) (x + 2)$

Step 3.3.1.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$16 x - 32 = x (x + 2) + 2 (x + 2)$

Step 3.3.1.2.2

Apply the distributive property.

$16 x - 32 = x \cdot x + x \cdot 2 + 2 (x + 2)$

Step 3.3.1.2.3

Apply the distributive property.

$16 x - 32 = x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2$

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 $x$ by $x$ .

$16 x - 32 = x^{2} + x \cdot 2 + 2 x + 2 \cdot 2$

Step 3.3.1.3.1.2

Move $2$ to the left of $x$ .

$16 x - 32 = x^{2} + 2 \cdot x + 2 x + 2 \cdot 2$

Step 3.3.1.3.1.3

Multiply $2$ by $2$ .

$16 x - 32 = x^{2} + 2 x + 2 x + 4$

Step 3.3.1.3.2

Add $2 x$ and $2 x$ .

$16 x - 32 = x^{2} + 4 x + 4$

Step 4

Solve for $x$ .

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Step 4.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 4 x + 4 = 16 x - 32$

Step 4.2

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

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Step 4.2.1

Subtract $16 x$ from both sides of the equation.

$x^{2} + 4 x + 4 - 16 x = - 32$

Step 4.2.2

Subtract $16 x$ from $4 x$ .

$x^{2} - 12 x + 4 = - 32$

Step 4.3

Add $32$ to both sides of the equation.

$x^{2} - 12 x + 4 + 32 = 0$

Step 4.4

Add $4$ and $32$ .

$x^{2} - 12 x + 36 = 0$

Step 4.5

Factor using the perfect square rule.

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Step 4.5.1

Rewrite $36$ as $6^{2}$ .

$x^{2} - 12 x + 6^{2} = 0$

Step 4.5.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$12 x = 2 \cdot x \cdot 6$

Step 4.5.3

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 6 + 6^{2} = 0$

Step 4.5.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 6$ .

${(x - 6)}^{2} = 0$

Step 4.6

Set the $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 4.7

Add $6$ to both sides of the equation.

$x = 6$