Algebra Examples

$\sqrt{x (x + 1)} = 0$

Step 1

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

${\sqrt{x (x + 1)}}^{2} = 0^{2}$

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

${({(x (x + 1))}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

${(x (x + 1))}^{\frac{1}{2} \cdot 2} = 0^{2}$

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.

${(x (x + 1))}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x (x + 1))}^{1} = 0^{2}$

Step 2.2.1.2

Apply the distributive property.

${(x \cdot x + x \cdot 1)}^{1} = 0^{2}$

Step 2.2.1.3

Simplify the expression.

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Step 2.2.1.3.1

Multiply $x$ by $x$ .

${(x^{2} + x \cdot 1)}^{1} = 0^{2}$

Step 2.2.1.3.2

Multiply $x$ by $1$ .

${(x^{2} + x)}^{1} = 0^{2}$

Step 2.2.1.3.3

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Raising $0$ to any positive power yields $0$ .

$x^{2} + x = 0$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

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

$x \cdot x + x = 0$

Step 3.1.2

Raise $x$ to the power of $1$ .

$x \cdot x + x = 0$

Step 3.1.3

Factor $x$ out of $x^{1}$ .

$x \cdot x + x \cdot 1 = 0$

Step 3.1.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

$x (x + 1) = 0$

Step 3.2

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 3.3

Set $x$ equal to $0$ .

$x = 0$

Step 3.4

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

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Step 3.4.1

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

$x + 1 = 0$

Step 3.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 3.5

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

$x = 0, - 1$