Solve Using the Quadratic Formula x^2-6x+45=0

$x^{2} - 6 x + 45 = 0$

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 $a = 1$ , $b = - 6$ , and $c = 45$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot 45)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 45}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 45$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 45}}{2 \cdot 1}$

Multiply $- 4$ by $45$ .

$x = \frac{6 \pm \sqrt{36 - 180}}{2 \cdot 1}$

Subtract $180$ from $36$ .

$x = \frac{6 \pm \sqrt{- 144}}{2 \cdot 1}$

Rewrite $- 144$ as $- 1 (144)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 144}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (144)}$ as $\sqrt{- 1} \cdot \sqrt{144}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{144}}{2 \cdot 1}$

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

$x = \frac{6 \pm i \cdot \sqrt{144}}{2 \cdot 1}$

Rewrite $144$ as $12^{2}$ .

$x = \frac{6 \pm i \cdot \sqrt{12^{2}}}{2 \cdot 1}$

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

$x = \frac{6 \pm i \cdot 12}{2 \cdot 1}$

Move $12$ to the left of $i$ .

$x = \frac{6 \pm 12 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 12 i}{2}$

Simplify $\frac{6 \pm 12 i}{2}$ .

$x = 3 \pm 6 i$