Finite Math Examples

$5 {(2 + x)}^{2} + 3 (2 + x) + 3$

Step 1

Set $5 {(2 + x)}^{2} + 3 (2 + x) + 3$ equal to $0$ .

$5 {(2 + x)}^{2} + 3 (2 + x) + 3 = 0$

Step 2

Simplify the left side.

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Step 2.1

Simplify $5 {(2 + x)}^{2} + 3 (2 + x) + 3$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$5 {(2 + x)}^{2} + 3 \cdot 2 + 3 x + 3 = 0$

Step 2.1.1.2

Multiply $3$ by $2$ .

$5 {(2 + x)}^{2} + 6 + 3 x + 3 = 0$

Step 2.1.2

Add $6$ and $3$ .

$5 {(2 + x)}^{2} + 3 x + 9 = 0$

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 $a = 5$ , $b = 23$ , and $c = 29$ into the quadratic formula and solve for $x$ .

$\frac{- 23 \pm \sqrt{23^{2} - 4 \cdot (5 \cdot 29)}}{2 \cdot 5}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $23$ to the power of $2$ .

$x = \frac{- 23 \pm \sqrt{529 - 4 \cdot 5 \cdot 29}}{2 \cdot 5}$

Step 5.1.2

Multiply $- 4 \cdot 5 \cdot 29$ .

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Step 5.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 23 \pm \sqrt{529 - 20 \cdot 29}}{2 \cdot 5}$

Step 5.1.2.2

Multiply $- 20$ by $29$ .

$x = \frac{- 23 \pm \sqrt{529 - 580}}{2 \cdot 5}$

Step 5.1.3

Subtract $580$ from $529$ .

$x = \frac{- 23 \pm \sqrt{- 51}}{2 \cdot 5}$

Step 5.1.4

Rewrite $- 51$ as $- 1 (51)$ .

$x = \frac{- 23 \pm \sqrt{- 1 \cdot 51}}{2 \cdot 5}$

Step 5.1.5

Rewrite $\sqrt{- 1 (51)}$ as $\sqrt{- 1} \cdot \sqrt{51}$ .

$x = \frac{- 23 \pm \sqrt{- 1} \cdot \sqrt{51}}{2 \cdot 5}$

Step 5.1.6

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

$x = \frac{- 23 \pm i \sqrt{51}}{2 \cdot 5}$

Step 5.2

Multiply $2$ by $5$ .

$x = \frac{- 23 \pm i \sqrt{51}}{10}$