Find All Complex Solutions 4x^2-30x+45=0

$4 x^{2} - 30 x + 45 = 0$

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 $a = 4$ , $b = - 30$ , and $c = 45$ into the quadratic formula and solve for $x$ .

$\frac{30 \pm \sqrt{{(- 30)}^{2} - 4 \cdot (4 \cdot 45)}}{2 \cdot 4}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 30$ to the power of $2$ .

$x = \frac{30 \pm \sqrt{900 - 4 \cdot 4 \cdot 45}}{2 \cdot 4}$

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

Multiply $- 4$ by $4$ .

$x = \frac{30 \pm \sqrt{900 - 16 \cdot 45}}{2 \cdot 4}$

Multiply $- 16$ by $45$ .

$x = \frac{30 \pm \sqrt{900 - 720}}{2 \cdot 4}$

Subtract $720$ from $900$ .

$x = \frac{30 \pm \sqrt{180}}{2 \cdot 4}$

Rewrite $180$ as $6^{2} \cdot 5$ .

Factor $36$ out of $180$ .

$x = \frac{30 \pm \sqrt{36 (5)}}{2 \cdot 4}$

Rewrite $36$ as $6^{2}$ .

$x = \frac{30 \pm \sqrt{6^{2} \cdot 5}}{2 \cdot 4}$

Pull terms out from under the radical.

$x = \frac{30 \pm 6 \sqrt{5}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{30 \pm 6 \sqrt{5}}{8}$

Simplify $\frac{30 \pm 6 \sqrt{5}}{8}$ .

$x = \frac{15 \pm 3 \sqrt{5}}{4}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{15 + 3 \sqrt{5}}{4}, \frac{15 - 3 \sqrt{5}}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{15 + 3 \sqrt{5}}{4}, \frac{15 - 3 \sqrt{5}}{4}$

Decimal Form:

$x = 5.42705098 \dots, 2.07294901 \dots$