Algebra Examples

$- 5 + 2 x^{2} = - 6 x$

Step 1

Add $6 x$ to both sides of the equation.

$- 5 + 2 x^{2} + 6 x = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 2$ , $b = 6$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (2 \cdot - 5)}}{2 \cdot 2}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 2 \cdot - 5}}{2 \cdot 2}$

Step 4.1.2

Multiply $- 4 \cdot 2 \cdot - 5$ .

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 8 \cdot - 5}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $- 8$ by $- 5$ .

$x = \frac{- 6 \pm \sqrt{36 + 40}}{2 \cdot 2}$

Step 4.1.3

Add $36$ and $40$ .

$x = \frac{- 6 \pm \sqrt{76}}{2 \cdot 2}$

Step 4.1.4

Rewrite $76$ as $2^{2} \cdot 19$ .

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Step 4.1.4.1

Factor $4$ out of $76$ .

$x = \frac{- 6 \pm \sqrt{4 (19)}}{2 \cdot 2}$

Step 4.1.4.2

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

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 2}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{19}}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$x = \frac{- 6 \pm 2 \sqrt{19}}{4}$

Step 4.3

Simplify $\frac{- 6 \pm 2 \sqrt{19}}{4}$ .

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

Step 5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 2 \cdot - 5}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot - 5$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 8 \cdot - 5}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $- 5$ .

$x = \frac{- 6 \pm \sqrt{36 + 40}}{2 \cdot 2}$

Step 5.1.3

Add $36$ and $40$ .

$x = \frac{- 6 \pm \sqrt{76}}{2 \cdot 2}$

Step 5.1.4

Rewrite $76$ as $2^{2} \cdot 19$ .

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Step 5.1.4.1

Factor $4$ out of $76$ .

$x = \frac{- 6 \pm \sqrt{4 (19)}}{2 \cdot 2}$

Step 5.1.4.2

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

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 2}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{19}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$x = \frac{- 6 \pm 2 \sqrt{19}}{4}$

Step 5.3

Simplify $\frac{- 6 \pm 2 \sqrt{19}}{4}$ .

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

Step 5.4

Change the $\pm$ to $+$ .

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

Step 5.5

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + \sqrt{19}}{2}$

Step 5.6

Factor $- 1$ out of $\sqrt{19}$ .

$x = \frac{- 1 \cdot 3 - 1 (- \sqrt{19})}{2}$

Step 5.7

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{19})$ .

$x = \frac{- 1 (3 - \sqrt{19})}{2}$

Step 5.8

Move the negative in front of the fraction.

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

Step 6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 2 \cdot - 5}}{2 \cdot 2}$

Step 6.1.2

Multiply $- 4 \cdot 2 \cdot - 5$ .

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Step 6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 8 \cdot - 5}}{2 \cdot 2}$

Step 6.1.2.2

Multiply $- 8$ by $- 5$ .

$x = \frac{- 6 \pm \sqrt{36 + 40}}{2 \cdot 2}$

Step 6.1.3

Add $36$ and $40$ .

$x = \frac{- 6 \pm \sqrt{76}}{2 \cdot 2}$

Step 6.1.4

Rewrite $76$ as $2^{2} \cdot 19$ .

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Step 6.1.4.1

Factor $4$ out of $76$ .

$x = \frac{- 6 \pm \sqrt{4 (19)}}{2 \cdot 2}$

Step 6.1.4.2

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

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 2}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{19}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$x = \frac{- 6 \pm 2 \sqrt{19}}{4}$

Step 6.3

Simplify $\frac{- 6 \pm 2 \sqrt{19}}{4}$ .

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

Step 6.4

Change the $\pm$ to $-$ .

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

Step 6.5

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - \sqrt{19}}{2}$

Step 6.6

Factor $- 1$ out of $- \sqrt{19}$ .

$x = \frac{- 1 \cdot 3 - (\sqrt{19})}{2}$

Step 6.7

Factor $- 1$ out of $- 1 (3) - (\sqrt{19})$ .

$x = \frac{- 1 (3 + \sqrt{19})}{2}$

Step 6.8

Move the negative in front of the fraction.

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

Step 7

The final answer is the combination of both solutions.

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.67944947 \dots, - 3.67944947 \dots$