Trigonometry Examples

$11.38 = - 0.5 x^{2} + x + 12$

Step 1

Rewrite the equation as $- 0.5 x^{2} + x + 12 = 11.38$ .

$- 0.5 x^{2} + x + 12 = 11.38$

Step 2

Subtract $11.38$ from both sides of the equation.

$- 0.5 x^{2} + x + 12 - 11.38 = 0$

Step 3

Subtract $11.38$ from $12$ .

$- 0.5 x^{2} + x + 0.62 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $- 1$ out of $- 0.5 x^{2} + x + 0.62$ .

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Step 4.1.1

Factor $- 1$ out of $- 0.5 x^{2}$ .

$- (0.5 x^{2}) + x + 0.62 = 0$

Step 4.1.2

Factor $- 1$ out of $x$ .

$- (0.5 x^{2}) - 1 (- x) + 0.62 = 0$

Step 4.1.3

Rewrite $0.62$ as $- 1 (- 0.62)$ .

$- (0.5 x^{2}) - 1 (- x) - 1 \cdot - 0.62 = 0$

Step 4.1.4

Factor $- 1$ out of $- (0.5 x^{2}) - 1 (- x)$ .

$- (0.5 x^{2} - x) - 1 \cdot - 0.62 = 0$

Step 4.1.5

Factor $- 1$ out of $- (0.5 x^{2} - x) - 1 (- 0.62)$ .

$- (0.5 x^{2} - x - 0.62) = 0$

Step 4.2

Factor.

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Step 4.2.1

Factor $0.02$ out of $0.5 x^{2} - x - 0.62$ .

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Step 4.2.1.1

Factor $0.02$ out of $0.5 x^{2}$ .

$- (0.02 (25 x^{2}) - x - 0.62) = 0$

Step 4.2.1.2

Factor $0.02$ out of $- x$ .

$- (0.02 (25 x^{2}) + 0.02 (- 50 x) - 0.62) = 0$

Step 4.2.1.3

Factor $0.02$ out of $- 0.62$ .

$- (0.02 (25 x^{2}) + 0.02 (- 50 x) + 0.02 (- 31)) = 0$

Step 4.2.1.4

Factor $0.02$ out of $0.02 (25 x^{2}) + 0.02 (- 50 x)$ .

$- (0.02 (25 x^{2} - 50 x) + 0.02 (- 31)) = 0$

Step 4.2.1.5

Factor $0.02$ out of $0.02 (25 x^{2} - 50 x) + 0.02 (- 31)$ .

$- (0.02 (25 x^{2} - 50 x - 31)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$- 1 \cdot (0.02 (25 x^{2} - 50 x - 31)) = 0$

Step 4.3

Multiply $- 1$ by $0.02$ .

$- 0.02 (25 x^{2} - 50 x - 31) = 0$

Step 5

Divide each term in $- 0.02 (25 x^{2} - 50 x - 31) = 0$ by $- 0.02$ and simplify.

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Step 5.1

Divide each term in $- 0.02 (25 x^{2} - 50 x - 31) = 0$ by $- 0.02$ .

$\frac{- 0.02 (25 x^{2} - 50 x - 31)}{- 0.02} = \frac{0}{- 0.02}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 0.02$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 0.02 (25 x^{2} - 50 x - 31)}{- 0.02} = \frac{0}{- 0.02}$

Step 5.2.1.2

Divide $25 x^{2} - 50 x - 31$ by $1$ .

$25 x^{2} - 50 x - 31 = \frac{0}{- 0.02}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $- 0.02$ .

$25 x^{2} - 50 x - 31 = 0$

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

Substitute the values $a = 25$ , $b = - 50$ , and $c = - 31$ into the quadratic formula and solve for $x$ .

$\frac{50 \pm \sqrt{{(- 50)}^{2} - 4 \cdot (25 \cdot - 31)}}{2 \cdot 25}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 50$ to the power of $2$ .

$x = \frac{50 \pm \sqrt{2500 - 4 \cdot 25 \cdot - 31}}{2 \cdot 25}$

Step 8.1.2

Multiply $- 4 \cdot 25 \cdot - 31$ .

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Step 8.1.2.1

Multiply $- 4$ by $25$ .

$x = \frac{50 \pm \sqrt{2500 - 100 \cdot - 31}}{2 \cdot 25}$

Step 8.1.2.2

Multiply $- 100$ by $- 31$ .

$x = \frac{50 \pm \sqrt{2500 + 3100}}{2 \cdot 25}$

Step 8.1.3

Add $2500$ and $3100$ .

$x = \frac{50 \pm \sqrt{5600}}{2 \cdot 25}$

Step 8.1.4

Rewrite $5600$ as $20^{2} \cdot 14$ .

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Step 8.1.4.1

Factor $400$ out of $5600$ .

$x = \frac{50 \pm \sqrt{400 (14)}}{2 \cdot 25}$

Step 8.1.4.2

Rewrite $400$ as $20^{2}$ .

$x = \frac{50 \pm \sqrt{20^{2} \cdot 14}}{2 \cdot 25}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{50 \pm 20 \sqrt{14}}{2 \cdot 25}$

Step 8.2

Multiply $2$ by $25$ .

$x = \frac{50 \pm 20 \sqrt{14}}{50}$

Step 8.3

Simplify $\frac{50 \pm 20 \sqrt{14}}{50}$ .

$x = \frac{5 \pm 2 \sqrt{14}}{5}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{5 + 2 \sqrt{14}}{5}, \frac{5 - 2 \sqrt{14}}{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 + 2 \sqrt{14}}{5}, \frac{5 - 2 \sqrt{14}}{5}$

Decimal Form:

$x = 2.49666295 \dots, - 0.49666295 \dots$