Find All Complex Solutions (3x+1)^2-100=0

${(3 x + 1)}^{2} - 100 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 9$ , $b = 6$ , and $c = - 99$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (9 \cdot - 99)}}{2 \cdot 9}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 9 \cdot - 99}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot - 99$ .

Multiply $- 4$ by $9$ .

$x = \frac{- 6 \pm \sqrt{36 - 36 \cdot - 99}}{2 \cdot 9}$

Multiply $- 36$ by $- 99$ .

$x = \frac{- 6 \pm \sqrt{36 + 3564}}{2 \cdot 9}$

Add $36$ and $3564$ .

$x = \frac{- 6 \pm \sqrt{3600}}{2 \cdot 9}$

Rewrite $3600$ as $60^{2}$ .

$x = \frac{- 6 \pm \sqrt{60^{2}}}{2 \cdot 9}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 6 \pm 60}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{- 6 \pm 60}{18}$

Simplify $\frac{- 6 \pm 60}{18}$ .

$x = \frac{- 1 \pm 10}{3}$

Step 4

The final answer is the combination of both solutions.

$x = 3, - \frac{11}{3}$