Algebra Examples

6 x^{2} - 6 x - 2 = 0

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

Step 1

Use the quadratic formula to find the solutions.

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

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

Step 2

Substitute the values

a = 6

$a = 6$ ,

b = - 6

$b = - 6$ , and

c = - 2

$c = - 2$ into the quadratic formula and solve for

x

$x$ .

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

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise

- 6

$- 6$ to the power of

2

$2$ .

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

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

Step 3.1.2

Multiply

- 4 \cdot 6 \cdot - 2

$- 4 \cdot 6 \cdot - 2$ .

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Step 3.1.2.1

Multiply

- 4

$- 4$ by

6

$6$ .

x = \frac{6 \pm \sqrt{36 - 24 \cdot - 2}}{2 \cdot 6}

$x = \frac{6 \pm \sqrt{36 - 24 \cdot - 2}}{2 \cdot 6}$

Step 3.1.2.2

Multiply

- 24

$- 24$ by

- 2

$- 2$ .

x = \frac{6 \pm \sqrt{36 + 48}}{2 \cdot 6}

$x = \frac{6 \pm \sqrt{36 + 48}}{2 \cdot 6}$

x = \frac{6 \pm \sqrt{36 + 48}}{2 \cdot 6}

$x = \frac{6 \pm \sqrt{36 + 48}}{2 \cdot 6}$

Step 3.1.3

Add

36

$36$ and

48

$48$ .

x = \frac{6 \pm \sqrt{84}}{2 \cdot 6}

$x = \frac{6 \pm \sqrt{84}}{2 \cdot 6}$

Step 3.1.4

Rewrite

84

$84$ as

2^{2} \cdot 21

$2^{2} \cdot 21$ .

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Step 3.1.4.1

Factor

4

$4$ out of

84

$84$ .

x = \frac{6 \pm \sqrt{4 (21)}}{2 \cdot 6}

$x = \frac{6 \pm \sqrt{4 (21)}}{2 \cdot 6}$

Step 3.1.4.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

x = \frac{6 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 6}

$x = \frac{6 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 6}$

x = \frac{6 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 6}

$x = \frac{6 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 6}$

Step 3.1.5

Pull terms out from under the radical.

x = \frac{6 \pm 2 \sqrt{21}}{2 \cdot 6}

$x = \frac{6 \pm 2 \sqrt{21}}{2 \cdot 6}$

x = \frac{6 \pm 2 \sqrt{21}}{2 \cdot 6}

$x = \frac{6 \pm 2 \sqrt{21}}{2 \cdot 6}$

Step 3.2

Multiply

2

$2$ by

6

$6$ .

x = \frac{6 \pm 2 \sqrt{21}}{12}

$x = \frac{6 \pm 2 \sqrt{21}}{12}$

Step 3.3

Simplify

\frac{6 \pm 2 \sqrt{21}}{12}

$\frac{6 \pm 2 \sqrt{21}}{12}$ .

x = \frac{3 \pm \sqrt{21}}{6}

$x = \frac{3 \pm \sqrt{21}}{6}$

x = \frac{3 \pm \sqrt{21}}{6}

$x = \frac{3 \pm \sqrt{21}}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

x = \frac{3 \pm \sqrt{21}}{6}

$x = \frac{3 \pm \sqrt{21}}{6}$

Decimal Form:

x = 1.26376261 \dots, - 0.26376261 \dots

$x = 1.26376261 \dots, - 0.26376261 \dots$