Precalculus Examples

$3 x^{2} - 8 x - 20 = 0$

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 $a = 3$ , $b = - 8$ , and $c = - 20$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (3 \cdot - 20)}}{2 \cdot 3}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 3 \cdot - 20}}{2 \cdot 3}$

Step 3.1.2

Multiply $- 4 \cdot 3 \cdot - 20$ .

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Step 3.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 12 \cdot - 20}}{2 \cdot 3}$

Step 3.1.2.2

Multiply $- 12$ by $- 20$ .

$x = \frac{8 \pm \sqrt{64 + 240}}{2 \cdot 3}$

Step 3.1.3

Add $64$ and $240$ .

$x = \frac{8 \pm \sqrt{304}}{2 \cdot 3}$

Step 3.1.4

Rewrite $304$ as $4^{2} \cdot 19$ .

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Step 3.1.4.1

Factor $16$ out of $304$ .

$x = \frac{8 \pm \sqrt{16 (19)}}{2 \cdot 3}$

Step 3.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{8 \pm \sqrt{4^{2} \cdot 19}}{2 \cdot 3}$

Step 3.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 4 \sqrt{19}}{2 \cdot 3}$

Step 3.2

Multiply $2$ by $3$ .

$x = \frac{8 \pm 4 \sqrt{19}}{6}$

Step 3.3

Simplify $\frac{8 \pm 4 \sqrt{19}}{6}$ .

$x = \frac{4 \pm 2 \sqrt{19}}{3}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{4 + 2 \sqrt{19}}{3}, \frac{4 - 2 \sqrt{19}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4 + 2 \sqrt{19}}{3}, \frac{4 - 2 \sqrt{19}}{3}$

Decimal Form:

$x = 4.23926596 \dots, - 1.57259929 \dots$