Precalculus Examples

$- 3 x^{2} - 6 x + 1 = 0$

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 $a = - 3$ , $b = - 6$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (- 3 \cdot 1)}}{2 \cdot - 3}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 3 \cdot 1}}{2 \cdot - 3}$

Step 3.1.2

Multiply $- 4 \cdot - 3 \cdot 1$ .

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Step 3.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{6 \pm \sqrt{36 + 12 \cdot 1}}{2 \cdot - 3}$

Step 3.1.2.2

Multiply $12$ by $1$ .

$x = \frac{6 \pm \sqrt{36 + 12}}{2 \cdot - 3}$

Step 3.1.3

Add $36$ and $12$ .

$x = \frac{6 \pm \sqrt{48}}{2 \cdot - 3}$

Step 3.1.4

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 3.1.4.1

Factor $16$ out of $48$ .

$x = \frac{6 \pm \sqrt{16 (3)}}{2 \cdot - 3}$

Step 3.1.4.2

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

$x = \frac{6 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot - 3}$

Step 3.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 4 \sqrt{3}}{2 \cdot - 3}$

Step 3.2

Multiply $2$ by $- 3$ .

$x = \frac{6 \pm 4 \sqrt{3}}{- 6}$

Step 3.3

Simplify $\frac{6 \pm 4 \sqrt{3}}{- 6}$ .

$x = \frac{3 \pm 2 \sqrt{3}}{- 3}$

Step 3.4

Move the negative in front of the fraction.

$x = - \frac{3 \pm 2 \sqrt{3}}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 \pm 2 \sqrt{3}}{3}$

Decimal Form:

$x = - 2.15470053 \dots, 0.15470053 \dots$