Trigonometry Examples

$f (x) = x^{2} + 2 x - 9$ , $f (x) = (x + 9)$

Step 1

Substitute $x + 9$ for $f (x)$ .

$x + 9 = x^{2} + 2 x - 9$

Step 2

Solve for $x$ .

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Step 2.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 2 x - 9 = x + 9$

Step 2.2

Move all terms containing $x$ to the left side of the equation.

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Step 2.2.1

Subtract $x$ from both sides of the equation.

$x^{2} + 2 x - 9 - x = 9$

Step 2.2.2

Subtract $x$ from $2 x$ .

$x^{2} + x - 9 = 9$

Step 2.3

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

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Step 2.3.1

Subtract $9$ from both sides of the equation.

$x^{2} + x - 9 - 9 = 0$

Step 2.3.2

Subtract $9$ from $- 9$ .

$x^{2} + x - 18 = 0$

Step 2.4

Use the quadratic formula to find the solutions.

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

Step 2.5

Substitute the values $a = 1$ , $b = 1$ , and $c = - 18$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot - 18)}}{2 \cdot 1}$

Step 2.6

Simplify.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 18}}{2 \cdot 1}$

Step 2.6.1.2

Multiply $- 4 \cdot 1 \cdot - 18$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 18}}{2 \cdot 1}$

Step 2.6.1.2.2

Multiply $- 4$ by $- 18$ .

$x = \frac{- 1 \pm \sqrt{1 + 72}}{2 \cdot 1}$

Step 2.6.1.3

Add $1$ and $72$ .

$x = \frac{- 1 \pm \sqrt{73}}{2 \cdot 1}$

Step 2.6.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{73}}{2}$

Step 2.7

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

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 18}}{2 \cdot 1}$

Step 2.7.1.2

Multiply $- 4 \cdot 1 \cdot - 18$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 18}}{2 \cdot 1}$

Step 2.7.1.2.2

Multiply $- 4$ by $- 18$ .

$x = \frac{- 1 \pm \sqrt{1 + 72}}{2 \cdot 1}$

Step 2.7.1.3

Add $1$ and $72$ .

$x = \frac{- 1 \pm \sqrt{73}}{2 \cdot 1}$

Step 2.7.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{73}}{2}$

Step 2.7.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{73}}{2}$

Step 2.7.4

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

$x = \frac{- 1 \cdot 1 + \sqrt{73}}{2}$

Step 2.7.5

Factor $- 1$ out of $\sqrt{73}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{73})}{2}$

Step 2.7.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{73})$ .

$x = \frac{- 1 (1 - \sqrt{73})}{2}$

Step 2.7.7

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{73}}{2}$

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

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 18}}{2 \cdot 1}$

Step 2.8.1.2

Multiply $- 4 \cdot 1 \cdot - 18$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 18}}{2 \cdot 1}$

Step 2.8.1.2.2

Multiply $- 4$ by $- 18$ .

$x = \frac{- 1 \pm \sqrt{1 + 72}}{2 \cdot 1}$

Step 2.8.1.3

Add $1$ and $72$ .

$x = \frac{- 1 \pm \sqrt{73}}{2 \cdot 1}$

Step 2.8.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{73}}{2}$

Step 2.8.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{73}}{2}$

Step 2.8.4

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

$x = \frac{- 1 \cdot 1 - \sqrt{73}}{2}$

Step 2.8.5

Factor $- 1$ out of $- \sqrt{73}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{73})}{2}$

Step 2.8.6

Factor $- 1$ out of $- 1 (1) - (\sqrt{73})$ .

$x = \frac{- 1 (1 + \sqrt{73})}{2}$

Step 2.8.7

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{73}}{2}$

Step 2.9

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{73}}{2}, - \frac{1 + \sqrt{73}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 - \sqrt{73}}{2}, - \frac{1 + \sqrt{73}}{2}$

Decimal Form:

$x = 3.77200187 \dots, - 4.77200187 \dots$