Trigonometry Examples

$f (x) = - \frac{1}{9} x^{2} + 25$ , $f (x) = 69 - 3 x$

Step 1

Substitute $69 - 3 x$ for $f (x)$ .

$69 - 3 x = - \frac{1}{9} x^{2} + 25$

Step 2

Solve for $x$ .

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Step 2.1

Combine $x^{2}$ and $\frac{1}{9}$ .

$69 - 3 x = - \frac{x^{2}}{9} + 25$

Step 2.2

Add $\frac{x^{2}}{9}$ to both sides of the equation.

$69 - 3 x + \frac{x^{2}}{9} = 25$

Step 2.3

Move all terms to the left side of the equation and simplify.

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Step 2.3.1

Subtract $25$ from both sides of the equation.

$69 - 3 x + \frac{x^{2}}{9} - 25 = 0$

Step 2.3.2

Subtract $25$ from $69$ .

$- 3 x + \frac{x^{2}}{9} + 44 = 0$

Step 2.4

Multiply through by the least common denominator $9$ , then simplify.

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Step 2.4.1

Apply the distributive property.

$9 (- 3 x) + 9 (\frac{x^{2}}{9}) + 9 \cdot 44 = 0$

Step 2.4.2

Simplify.

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Step 2.4.2.1

Multiply $- 3$ by $9$ .

$- 27 x + 9 (\frac{x^{2}}{9}) + 9 \cdot 44 = 0$

Step 2.4.2.2

Cancel the common factor of $9$ .

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Step 2.4.2.2.1

Cancel the common factor.

$- 27 x + 9 (\frac{x^{2}}{9}) + 9 \cdot 44 = 0$

Step 2.4.2.2.2

Rewrite the expression.

$- 27 x + x^{2} + 9 \cdot 44 = 0$

Step 2.4.2.3

Multiply $9$ by $44$ .

$- 27 x + x^{2} + 396 = 0$

Step 2.4.3

Reorder $- 27 x$ and $x^{2}$ .

$x^{2} - 27 x + 396 = 0$

Step 2.5

Use the quadratic formula to find the solutions.

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

Step 2.6

Substitute the values $a = 1$ , $b = - 27$ , and $c = 396$ into the quadratic formula and solve for $x$ .

$\frac{27 \pm \sqrt{{(- 27)}^{2} - 4 \cdot (1 \cdot 396)}}{2 \cdot 1}$

Step 2.7

Simplify.

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $- 27$ to the power of $2$ .

$x = \frac{27 \pm \sqrt{729 - 4 \cdot 1 \cdot 396}}{2 \cdot 1}$

Step 2.7.1.2

Multiply $- 4 \cdot 1 \cdot 396$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{27 \pm \sqrt{729 - 4 \cdot 396}}{2 \cdot 1}$

Step 2.7.1.2.2

Multiply $- 4$ by $396$ .

$x = \frac{27 \pm \sqrt{729 - 1584}}{2 \cdot 1}$

Step 2.7.1.3

Subtract $1584$ from $729$ .

$x = \frac{27 \pm \sqrt{- 855}}{2 \cdot 1}$

Step 2.7.1.4

Rewrite $- 855$ as $- 1 (855)$ .

$x = \frac{27 \pm \sqrt{- 1 \cdot 855}}{2 \cdot 1}$

Step 2.7.1.5

Rewrite $\sqrt{- 1 (855)}$ as $\sqrt{- 1} \cdot \sqrt{855}$ .

$x = \frac{27 \pm \sqrt{- 1} \cdot \sqrt{855}}{2 \cdot 1}$

Step 2.7.1.6

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

$x = \frac{27 \pm i \cdot \sqrt{855}}{2 \cdot 1}$

Step 2.7.1.7

Rewrite $855$ as $3^{2} \cdot 95$ .

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Step 2.7.1.7.1

Factor $9$ out of $855$ .

$x = \frac{27 \pm i \cdot \sqrt{9 (95)}}{2 \cdot 1}$

Step 2.7.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{27 \pm i \cdot \sqrt{3^{2} \cdot 95}}{2 \cdot 1}$

Step 2.7.1.8

Pull terms out from under the radical.

$x = \frac{27 \pm i \cdot (3 \sqrt{95})}{2 \cdot 1}$

Step 2.7.1.9

Move $3$ to the left of $i$ .

$x = \frac{27 \pm 3 i \sqrt{95}}{2 \cdot 1}$

Step 2.7.2

Multiply $2$ by $1$ .

$x = \frac{27 \pm 3 i \sqrt{95}}{2}$

Step 2.8

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

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

Raise $- 27$ to the power of $2$ .

$x = \frac{27 \pm \sqrt{729 - 4 \cdot 1 \cdot 396}}{2 \cdot 1}$

Step 2.8.1.2

Multiply $- 4 \cdot 1 \cdot 396$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{27 \pm \sqrt{729 - 4 \cdot 396}}{2 \cdot 1}$

Step 2.8.1.2.2

Multiply $- 4$ by $396$ .

$x = \frac{27 \pm \sqrt{729 - 1584}}{2 \cdot 1}$

Step 2.8.1.3

Subtract $1584$ from $729$ .

$x = \frac{27 \pm \sqrt{- 855}}{2 \cdot 1}$

Step 2.8.1.4

Rewrite $- 855$ as $- 1 (855)$ .

$x = \frac{27 \pm \sqrt{- 1 \cdot 855}}{2 \cdot 1}$

Step 2.8.1.5

Rewrite $\sqrt{- 1 (855)}$ as $\sqrt{- 1} \cdot \sqrt{855}$ .

$x = \frac{27 \pm \sqrt{- 1} \cdot \sqrt{855}}{2 \cdot 1}$

Step 2.8.1.6

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

$x = \frac{27 \pm i \cdot \sqrt{855}}{2 \cdot 1}$

Step 2.8.1.7

Rewrite $855$ as $3^{2} \cdot 95$ .

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Step 2.8.1.7.1

Factor $9$ out of $855$ .

$x = \frac{27 \pm i \cdot \sqrt{9 (95)}}{2 \cdot 1}$

Step 2.8.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{27 \pm i \cdot \sqrt{3^{2} \cdot 95}}{2 \cdot 1}$

Step 2.8.1.8

Pull terms out from under the radical.

$x = \frac{27 \pm i \cdot (3 \sqrt{95})}{2 \cdot 1}$

Step 2.8.1.9

Move $3$ to the left of $i$ .

$x = \frac{27 \pm 3 i \sqrt{95}}{2 \cdot 1}$

Step 2.8.2

Multiply $2$ by $1$ .

$x = \frac{27 \pm 3 i \sqrt{95}}{2}$

Step 2.8.3

Change the $\pm$ to $+$ .

$x = \frac{27 + 3 i \sqrt{95}}{2}$

Step 2.9

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

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Step 2.9.1

Simplify the numerator.

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Step 2.9.1.1

Raise $- 27$ to the power of $2$ .

$x = \frac{27 \pm \sqrt{729 - 4 \cdot 1 \cdot 396}}{2 \cdot 1}$

Step 2.9.1.2

Multiply $- 4 \cdot 1 \cdot 396$ .

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Step 2.9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{27 \pm \sqrt{729 - 4 \cdot 396}}{2 \cdot 1}$

Step 2.9.1.2.2

Multiply $- 4$ by $396$ .

$x = \frac{27 \pm \sqrt{729 - 1584}}{2 \cdot 1}$

Step 2.9.1.3

Subtract $1584$ from $729$ .

$x = \frac{27 \pm \sqrt{- 855}}{2 \cdot 1}$

Step 2.9.1.4

Rewrite $- 855$ as $- 1 (855)$ .

$x = \frac{27 \pm \sqrt{- 1 \cdot 855}}{2 \cdot 1}$

Step 2.9.1.5

Rewrite $\sqrt{- 1 (855)}$ as $\sqrt{- 1} \cdot \sqrt{855}$ .

$x = \frac{27 \pm \sqrt{- 1} \cdot \sqrt{855}}{2 \cdot 1}$

Step 2.9.1.6

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

$x = \frac{27 \pm i \cdot \sqrt{855}}{2 \cdot 1}$

Step 2.9.1.7

Rewrite $855$ as $3^{2} \cdot 95$ .

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Step 2.9.1.7.1

Factor $9$ out of $855$ .

$x = \frac{27 \pm i \cdot \sqrt{9 (95)}}{2 \cdot 1}$

Step 2.9.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{27 \pm i \cdot \sqrt{3^{2} \cdot 95}}{2 \cdot 1}$

Step 2.9.1.8

Pull terms out from under the radical.

$x = \frac{27 \pm i \cdot (3 \sqrt{95})}{2 \cdot 1}$

Step 2.9.1.9

Move $3$ to the left of $i$ .

$x = \frac{27 \pm 3 i \sqrt{95}}{2 \cdot 1}$

Step 2.9.2

Multiply $2$ by $1$ .

$x = \frac{27 \pm 3 i \sqrt{95}}{2}$

Step 2.9.3

Change the $\pm$ to $-$ .

$x = \frac{27 - 3 i \sqrt{95}}{2}$

Step 2.10

The final answer is the combination of both solutions.

$x = \frac{27 + 3 i \sqrt{95}}{2}, \frac{27 - 3 i \sqrt{95}}{2}$