Finite Math Examples

$2 x^{2} + 12 x = 10$

Step 1

Subtract $10$ from both sides of the equation.

$2 x^{2} + 12 x - 10 = 0$

Step 2

Factor $2$ out of $2 x^{2} + 12 x - 10$ .

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Step 2.1

Factor $2$ out of $2 x^{2}$ .

$2 (x^{2}) + 12 x - 10 = 0$

Step 2.2

Factor $2$ out of $12 x$ .

$2 (x^{2}) + 2 (6 x) - 10 = 0$

Step 2.3

Factor $2$ out of $- 10$ .

$2 x^{2} + 2 (6 x) + 2 \cdot - 5 = 0$

Step 2.4

Factor $2$ out of $2 x^{2} + 2 (6 x)$ .

$2 (x^{2} + 6 x) + 2 \cdot - 5 = 0$

Step 2.5

Factor $2$ out of $2 (x^{2} + 6 x) + 2 \cdot - 5$ .

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

Step 3

Divide each term in $2 (x^{2} + 6 x - 5) = 0$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 (x^{2} + 6 x - 5) = 0$ by $2$ .

$\frac{2 (x^{2} + 6 x - 5)}{2} = \frac{0}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} + 6 x - 5)}{2} = \frac{0}{2}$

Step 3.2.1.2

Divide $x^{2} + 6 x - 5$ by $1$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $2$ .

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

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 1$ , $b = 6$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot - 5)}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 5}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $- 5$ .

$x = \frac{- 6 \pm \sqrt{36 + 20}}{2 \cdot 1}$

Step 6.1.3

Add $36$ and $20$ .

$x = \frac{- 6 \pm \sqrt{56}}{2 \cdot 1}$

Step 6.1.4

Rewrite $56$ as $2^{2} \cdot 14$ .

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Step 6.1.4.1

Factor $4$ out of $56$ .

$x = \frac{- 6 \pm \sqrt{4 (14)}}{2 \cdot 1}$

Step 6.1.4.2

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

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 14}}{2 \cdot 1}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{14}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 2 \sqrt{14}}{2}$

Step 6.3

Simplify $\frac{- 6 \pm 2 \sqrt{14}}{2}$ .

$x = - 3 \pm \sqrt{14}$

Step 7

The final answer is the combination of both solutions.

$x = - 3 + \sqrt{14}, - 3 - \sqrt{14}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - 3 + \sqrt{14}, - 3 - \sqrt{14}$

Decimal Form:

$x = 0.74165738 \dots, - 6.74165738 \dots$