Trigonometry Examples

$f (x) = 3 - x^{2}$ , $f (x) = x^{2} + 2 x - 15$

Step 1

Substitute $x^{2} + 2 x - 15$ for $f (x)$ .

$x^{2} + 2 x - 15 = 3 - x^{2}$

Step 2

Solve for $x$ .

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Step 2.1

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

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Step 2.1.1

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

$x^{2} + 2 x - 15 + x^{2} = 3$

Step 2.1.2

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} + 2 x - 15 = 3$

Step 2.2

Subtract $3$ from both sides of the equation.

$2 x^{2} + 2 x - 15 - 3 = 0$

Step 2.3

Subtract $3$ from $- 15$ .

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

Step 2.4

Factor $2$ out of $2 x^{2} + 2 x - 18$ .

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Step 2.4.1

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

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

Step 2.4.2

Factor $2$ out of $2 x$ .

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

Step 2.4.3

Factor $2$ out of $- 18$ .

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

Step 2.4.4

Factor $2$ out of $2 (x^{2}) + 2 x$ .

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

Step 2.4.5

Factor $2$ out of $2 (x^{2} + x) + 2 \cdot - 9$ .

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

Step 2.5

Divide each term in $2 (x^{2} + x - 9) = 0$ by $2$ and simplify.

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Step 2.5.1

Divide each term in $2 (x^{2} + x - 9) = 0$ by $2$ .

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

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $2$ .

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Step 2.5.2.1.1

Cancel the common factor.

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

Step 2.5.2.1.2

Divide $x^{2} + x - 9$ by $1$ .

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

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Divide $0$ by $2$ .

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

Step 2.6

Use the quadratic formula to find the solutions.

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

Step 2.7

Substitute the values $a = 1$ , $b = 1$ , and $c = - 9$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot - 9)}}{2 \cdot 1}$

Step 2.8

Simplify.

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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.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Step 2.8.1.2

Multiply $- 4 \cdot 1 \cdot - 9$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 9}}{2 \cdot 1}$

Step 2.8.1.2.2

Multiply $- 4$ by $- 9$ .

$x = \frac{- 1 \pm \sqrt{1 + 36}}{2 \cdot 1}$

Step 2.8.1.3

Add $1$ and $36$ .

$x = \frac{- 1 \pm \sqrt{37}}{2 \cdot 1}$

Step 2.8.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{37}}{2}$

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

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Step 2.9.1.2

Multiply $- 4 \cdot 1 \cdot - 9$ .

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Step 2.9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 9}}{2 \cdot 1}$

Step 2.9.1.2.2

Multiply $- 4$ by $- 9$ .

$x = \frac{- 1 \pm \sqrt{1 + 36}}{2 \cdot 1}$

Step 2.9.1.3

Add $1$ and $36$ .

$x = \frac{- 1 \pm \sqrt{37}}{2 \cdot 1}$

Step 2.9.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{37}}{2}$

Step 2.9.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{37}}{2}$

Step 2.9.4

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

$x = \frac{- 1 \cdot 1 + \sqrt{37}}{2}$

Step 2.9.5

Factor $- 1$ out of $\sqrt{37}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{37})}{2}$

Step 2.9.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{37})$ .

$x = \frac{- 1 (1 - \sqrt{37})}{2}$

Step 2.9.7

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{37}}{2}$

Step 2.10

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

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Step 2.10.1

Simplify the numerator.

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Step 2.10.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Step 2.10.1.2

Multiply $- 4 \cdot 1 \cdot - 9$ .

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Step 2.10.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 9}}{2 \cdot 1}$

Step 2.10.1.2.2

Multiply $- 4$ by $- 9$ .

$x = \frac{- 1 \pm \sqrt{1 + 36}}{2 \cdot 1}$

Step 2.10.1.3

Add $1$ and $36$ .

$x = \frac{- 1 \pm \sqrt{37}}{2 \cdot 1}$

Step 2.10.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{37}}{2}$

Step 2.10.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{37}}{2}$

Step 2.10.4

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

$x = \frac{- 1 \cdot 1 - \sqrt{37}}{2}$

Step 2.10.5

Factor $- 1$ out of $- \sqrt{37}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{37})}{2}$

Step 2.10.6

Factor $- 1$ out of $- 1 (1) - (\sqrt{37})$ .

$x = \frac{- 1 (1 + \sqrt{37})}{2}$

Step 2.10.7

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{37}}{2}$

Step 2.11

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{37}}{2}, - \frac{1 + \sqrt{37}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 - \sqrt{37}}{2}, - \frac{1 + \sqrt{37}}{2}$

Decimal Form:

$x = 2.54138126 \dots, - 3.54138126 \dots$