Algebra Examples

$\sqrt{x^{2} + 3 x} = 2 x$

Step 1

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

${\sqrt{x^{2} + 3 x}}^{2} = {(2 x)}^{2}$

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 $\sqrt{x^{2} + 3 x}$ as ${(x^{2} + 3 x)}^{\frac{1}{2}}$ .

${({(x^{2} + 3 x)}^{\frac{1}{2}})}^{2} = {(2 x)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x^{2} + 3 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(x^{2} + 3 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(x^{2} + 3 x)}^{\frac{1}{2} \cdot 2} = {(2 x)}^{2}$

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.

${(x^{2} + 3 x)}^{\frac{1}{2} \cdot 2} = {(2 x)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x^{2} + 3 x)}^{1} = {(2 x)}^{2}$

Step 2.2.1.2

Simplify.

$x^{2} + 3 x = {(2 x)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the product rule to $2 x$ .

$x^{2} + 3 x = 2^{2} x^{2}$

Step 2.3.1.2

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

$x^{2} + 3 x = 4 x^{2}$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $4 x^{2}$ from both sides of the equation.

$x^{2} + 3 x - 4 x^{2} = 0$

Step 3.1.2

Subtract $4 x^{2}$ from $x^{2}$ .

$- 3 x^{2} + 3 x = 0$

Step 3.2

Factor $- 3 x$ out of $- 3 x^{2} + 3 x$ .

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Step 3.2.1

Factor $- 3 x$ out of $- 3 x^{2}$ .

$- 3 x \cdot x + 3 x = 0$

Step 3.2.2

Factor $- 3 x$ out of $3 x$ .

$- 3 x \cdot x - 3 x \cdot - 1 = 0$

Step 3.2.3

Factor $- 3 x$ out of $- 3 x (x) - 3 x (- 1)$ .

$- 3 x (x - 1) = 0$

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 $0$ .

$x = 0$

$x - 1 = 0$

Step 3.4

Set $x$ equal to $0$ .

$x = 0$

Step 3.5

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.6

The final solution is all the values that make $- 3 x (x - 1) = 0$ true.

$x = 0, 1$