Algebra Examples

$x + 3 = \sqrt{- 20 + 12 x + 3}$

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.

$\sqrt{- 20 + 12 x + 3} = x + 3$

Step 2

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

${\sqrt{- 20 + 12 x + 3}}^{2} = {(x + 3)}^{2}$

Step 3

Simplify each side of the equation.

Tap for more steps...

Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 20 + 12 x + 3}$ as ${(- 20 + 12 x + 3)}^{\frac{1}{2}}$ .

${({(- 20 + 12 x + 3)}^{\frac{1}{2}})}^{2} = {(x + 3)}^{2}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify ${({(- 20 + 12 x + 3)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.1

Multiply the exponents in ${({(- 20 + 12 x + 3)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.1.1

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

${(- 20 + 12 x + 3)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.1.2.1

Cancel the common factor.

${(- 20 + 12 x + 3)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(- 20 + 12 x + 3)}^{1} = {(x + 3)}^{2}$

Step 3.2.1.2

Add $- 20$ and $3$ .

${(12 x - 17)}^{1} = {(x + 3)}^{2}$

Step 3.2.1.3

Simplify.

$12 x - 17 = {(x + 3)}^{2}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

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

Tap for more steps...

Step 3.3.1.1

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$12 x - 17 = (x + 3) (x + 3)$

Step 3.3.1.2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Tap for more steps...

Step 3.3.1.2.1

Apply the distributive property.

$12 x - 17 = x (x + 3) + 3 (x + 3)$

Step 3.3.1.2.2

Apply the distributive property.

$12 x - 17 = x \cdot x + x \cdot 3 + 3 (x + 3)$

Step 3.3.1.2.3

Apply the distributive property.

$12 x - 17 = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3$

Step 3.3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 3.3.1.3.1

Simplify each term.

Tap for more steps...

Step 3.3.1.3.1.1

Multiply $x$ by $x$ .

$12 x - 17 = x^{2} + x \cdot 3 + 3 x + 3 \cdot 3$

Step 3.3.1.3.1.2

Move $3$ to the left of $x$ .

$12 x - 17 = x^{2} + 3 \cdot x + 3 x + 3 \cdot 3$

Step 3.3.1.3.1.3

Multiply $3$ by $3$ .

$12 x - 17 = x^{2} + 3 x + 3 x + 9$

Step 3.3.1.3.2

Add $3 x$ and $3 x$ .

$12 x - 17 = x^{2} + 6 x + 9$

Step 4

Solve for $x$ .

Tap for more steps...

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.

$x^{2} + 6 x + 9 = 12 x - 17$

Step 4.2

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

Tap for more steps...

Step 4.2.1

Subtract $12 x$ from both sides of the equation.

$x^{2} + 6 x + 9 - 12 x = - 17$

Step 4.2.2

Subtract $12 x$ from $6 x$ .

$x^{2} - 6 x + 9 = - 17$

Step 4.3

Move all terms to the left side of the equation and simplify.

Tap for more steps...

Step 4.3.1

Add $17$ to both sides of the equation.

$x^{2} - 6 x + 9 + 17 = 0$

Step 4.3.2

Add $9$ and $17$ .

$x^{2} - 6 x + 26 = 0$

Step 4.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.5

Substitute the values $a = 1$ , $b = - 6$ , and $c = 26$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot 26)}}{2 \cdot 1}$

Step 4.6

Simplify.

Tap for more steps...

Step 4.6.1

Simplify the numerator.

Tap for more steps...

Step 4.6.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 26}}{2 \cdot 1}$

Step 4.6.1.2

Multiply $- 4 \cdot 1 \cdot 26$ .

Tap for more steps...

Step 4.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 26}}{2 \cdot 1}$

Step 4.6.1.2.2

Multiply $- 4$ by $26$ .

$x = \frac{6 \pm \sqrt{36 - 104}}{2 \cdot 1}$

Step 4.6.1.3

Subtract $104$ from $36$ .

$x = \frac{6 \pm \sqrt{- 68}}{2 \cdot 1}$

Step 4.6.1.4

Rewrite $- 68$ as $- 1 (68)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 68}}{2 \cdot 1}$

Step 4.6.1.5

Rewrite $\sqrt{- 1 (68)}$ as $\sqrt{- 1} \cdot \sqrt{68}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{68}}{2 \cdot 1}$

Step 4.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{6 \pm i \cdot \sqrt{68}}{2 \cdot 1}$

Step 4.6.1.7

Rewrite $68$ as $2^{2} \cdot 17$ .

Tap for more steps...

Step 4.6.1.7.1

Factor $4$ out of $68$ .

$x = \frac{6 \pm i \cdot \sqrt{4 (17)}}{2 \cdot 1}$

Step 4.6.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{6 \pm i \cdot \sqrt{2^{2} \cdot 17}}{2 \cdot 1}$

Step 4.6.1.8

Pull terms out from under the radical.

$x = \frac{6 \pm i \cdot (2 \sqrt{17})}{2 \cdot 1}$

Step 4.6.1.9

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

$x = \frac{6 \pm 2 i \sqrt{17}}{2 \cdot 1}$

Step 4.6.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 i \sqrt{17}}{2}$

Step 4.6.3

Simplify $\frac{6 \pm 2 i \sqrt{17}}{2}$ .

$x = 3 \pm i \sqrt{17}$

Step 4.7

The final answer is the combination of both solutions.

$x = 3 + i \sqrt{17}, 3 - i \sqrt{17}$