Algebra Examples

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

Step 1

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

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

Step 2

Simplify each side of the equation.

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 3 x}$ as ${(x^{2} + 3 x)}^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

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

Tap for more steps...

Step 2.2.1.1

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

Tap for more steps...

Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

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

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

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Subtract $4$ from both sides of the equation.

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

Step 3.2

Factor $x^{2} + 3 x - 4$ using the AC method.

Tap for more steps...

Step 3.2.1

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 $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 3.2.2

Write the factored form using these integers.

$(x - 1) (x + 4) = 0$

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 $0$ .

$x - 1 = 0$

$x + 4 = 0$

Step 3.4

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

Tap for more steps...

Step 3.4.1

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

$x - 1 = 0$

Step 3.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.5

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

Tap for more steps...

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 $(x - 1) (x + 4) = 0$ true.

$x = 1, - 4$