Finite Math Examples

$5 x^{2} + 13 x + 2 (x + 6)$

Step 1

Set $5 x^{2} + 13 x + 2 (x + 6)$ equal to $0$ .

$5 x^{2} + 13 x + 2 (x + 6) = 0$

Step 2

Simplify the left side.

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Step 2.1

Simplify $5 x^{2} + 13 x + 2 (x + 6)$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$5 x^{2} + 13 x + 2 x + 2 \cdot 6 = 0$

Step 2.1.1.2

Multiply $2$ by $6$ .

$5 x^{2} + 13 x + 2 x + 12 = 0$

Step 2.1.2

Add $13 x$ and $2 x$ .

$5 x^{2} + 15 x + 12 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 5$ , $b = 15$ , and $c = 12$ into the quadratic formula and solve for $x$ .

$\frac{- 15 \pm \sqrt{15^{2} - 4 \cdot (5 \cdot 12)}}{2 \cdot 5}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $15$ to the power of $2$ .

$x = \frac{- 15 \pm \sqrt{225 - 4 \cdot 5 \cdot 12}}{2 \cdot 5}$

Step 5.1.2

Multiply $- 4 \cdot 5 \cdot 12$ .

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Step 5.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 15 \pm \sqrt{225 - 20 \cdot 12}}{2 \cdot 5}$

Step 5.1.2.2

Multiply $- 20$ by $12$ .

$x = \frac{- 15 \pm \sqrt{225 - 240}}{2 \cdot 5}$

Step 5.1.3

Subtract $240$ from $225$ .

$x = \frac{- 15 \pm \sqrt{- 15}}{2 \cdot 5}$

Step 5.1.4

Rewrite $- 15$ as $- 1 (15)$ .

$x = \frac{- 15 \pm \sqrt{- 1 \cdot 15}}{2 \cdot 5}$

Step 5.1.5

Rewrite $\sqrt{- 1 (15)}$ as $\sqrt{- 1} \cdot \sqrt{15}$ .

$x = \frac{- 15 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot 5}$

Step 5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 15 \pm i \sqrt{15}}{2 \cdot 5}$

Step 5.2

Multiply $2$ by $5$ .

$x = \frac{- 15 \pm i \sqrt{15}}{10}$