Algebra Examples

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

Step 1

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

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

Step 2

Simplify each side of the equation.

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Multiply the exponents in ${({(2 x^{2} + 4 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

$2 x^{2} + 4 x = 4^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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

Step 3

Solve for $x$ .

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Step 3.1

Subtract $16$ from both sides of the equation.

$2 x^{2} + 4 x - 16 = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $2$ out of $2 x^{2} + 4 x - 16$ .

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Step 3.2.1.1

Factor $2$ out of $2 x^{2}$ .

$2 (x^{2}) + 4 x - 16 = 0$

Step 3.2.1.2

Factor $2$ out of $4 x$ .

$2 (x^{2}) + 2 (2 x) - 16 = 0$

Step 3.2.1.3

Factor $2$ out of $- 16$ .

$2 x^{2} + 2 (2 x) + 2 \cdot - 8 = 0$

Step 3.2.1.4

Factor $2$ out of $2 x^{2} + 2 (2 x)$ .

$2 (x^{2} + 2 x) + 2 \cdot - 8 = 0$

Step 3.2.1.5

Factor $2$ out of $2 (x^{2} + 2 x) + 2 \cdot - 8$ .

$2 (x^{2} + 2 x - 8) = 0$

Step 3.2.2

Factor.

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Step 3.2.2.1

Factor $x^{2} + 2 x - 8$ using the AC method.

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Step 3.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 8$ and whose sum is $2$ .

$- 2, 4$

Step 3.2.2.1.2

Write the factored form using these integers.

$2 ((x - 2) (x + 4)) = 0$

Step 3.2.2.2

Remove unnecessary parentheses.

$2 (x - 2) (x + 4) = 0$

Step 3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$x + 4 = 0$

Step 3.4

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 3.4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.5

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 3.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 3.6

The final solution is all the values that make $2 (x - 2) (x + 4) = 0$ true.

$x = 2, - 4$