Trigonometry Examples

$y = 2 x^{2} + 6 x - 54$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 2 x^{2} + 6 x - 54$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 x^{2} + 6 x - 54 = 0$ .

$2 x^{2} + 6 x - 54 = 0$

Step 1.2.2

Factor $2$ out of $2 x^{2} + 6 x - 54$ .

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Step 1.2.2.1

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

$2 (x^{2}) + 6 x - 54 = 0$

Step 1.2.2.2

Factor $2$ out of $6 x$ .

$2 (x^{2}) + 2 (3 x) - 54 = 0$

Step 1.2.2.3

Factor $2$ out of $- 54$ .

$2 x^{2} + 2 (3 x) + 2 \cdot - 27 = 0$

Step 1.2.2.4

Factor $2$ out of $2 x^{2} + 2 (3 x)$ .

$2 (x^{2} + 3 x) + 2 \cdot - 27 = 0$

Step 1.2.2.5

Factor $2$ out of $2 (x^{2} + 3 x) + 2 \cdot - 27$ .

$2 (x^{2} + 3 x - 27) = 0$

Step 1.2.3

Divide each term in $2 (x^{2} + 3 x - 27) = 0$ by $2$ and simplify.

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Step 1.2.3.1

Divide each term in $2 (x^{2} + 3 x - 27) = 0$ by $2$ .

$\frac{2 (x^{2} + 3 x - 27)}{2} = \frac{0}{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} + 3 x - 27)}{2} = \frac{0}{2}$

Step 1.2.3.2.1.2

Divide $x^{2} + 3 x - 27$ by $1$ .

$x^{2} + 3 x - 27 = \frac{0}{2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $2$ .

$x^{2} + 3 x - 27 = 0$

Step 1.2.4

Use the quadratic formula to find the solutions.

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

Step 1.2.5

Substitute the values $a = 1$ , $b = 3$ , and $c = - 27$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot - 27)}}{2 \cdot 1}$

Step 1.2.6

Simplify.

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 27}}{2 \cdot 1}$

Step 1.2.6.1.2

Multiply $- 4 \cdot 1 \cdot - 27$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 27}}{2 \cdot 1}$

Step 1.2.6.1.2.2

Multiply $- 4$ by $- 27$ .

$x = \frac{- 3 \pm \sqrt{9 + 108}}{2 \cdot 1}$

Step 1.2.6.1.3

Add $9$ and $108$ .

$x = \frac{- 3 \pm \sqrt{117}}{2 \cdot 1}$

Step 1.2.6.1.4

Rewrite $117$ as $3^{2} \cdot 13$ .

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Step 1.2.6.1.4.1

Factor $9$ out of $117$ .

$x = \frac{- 3 \pm \sqrt{9 (13)}}{2 \cdot 1}$

Step 1.2.6.1.4.2

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

$x = \frac{- 3 \pm \sqrt{3^{2} \cdot 13}}{2 \cdot 1}$

Step 1.2.6.1.5

Pull terms out from under the radical.

$x = \frac{- 3 \pm 3 \sqrt{13}}{2 \cdot 1}$

Step 1.2.6.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm 3 \sqrt{13}}{2}$

Step 1.2.7

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

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Step 1.2.7.1

Simplify the numerator.

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Step 1.2.7.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 27}}{2 \cdot 1}$

Step 1.2.7.1.2

Multiply $- 4 \cdot 1 \cdot - 27$ .

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Step 1.2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 27}}{2 \cdot 1}$

Step 1.2.7.1.2.2

Multiply $- 4$ by $- 27$ .

$x = \frac{- 3 \pm \sqrt{9 + 108}}{2 \cdot 1}$

Step 1.2.7.1.3

Add $9$ and $108$ .

$x = \frac{- 3 \pm \sqrt{117}}{2 \cdot 1}$

Step 1.2.7.1.4

Rewrite $117$ as $3^{2} \cdot 13$ .

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Step 1.2.7.1.4.1

Factor $9$ out of $117$ .

$x = \frac{- 3 \pm \sqrt{9 (13)}}{2 \cdot 1}$

Step 1.2.7.1.4.2

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

$x = \frac{- 3 \pm \sqrt{3^{2} \cdot 13}}{2 \cdot 1}$

Step 1.2.7.1.5

Pull terms out from under the radical.

$x = \frac{- 3 \pm 3 \sqrt{13}}{2 \cdot 1}$

Step 1.2.7.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm 3 \sqrt{13}}{2}$

Step 1.2.7.3

Change the $\pm$ to $+$ .

$x = \frac{- 3 + 3 \sqrt{13}}{2}$

Step 1.2.7.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + 3 \sqrt{13}}{2}$

Step 1.2.7.5

Factor $- 1$ out of $3 \sqrt{13}$ .

$x = \frac{- 1 \cdot 3 - (- 3 \sqrt{13})}{2}$

Step 1.2.7.6

Factor $- 1$ out of $- 1 (3) - (- 3 \sqrt{13})$ .

$x = \frac{- 1 (3 - 3 \sqrt{13})}{2}$

Step 1.2.7.7

Move the negative in front of the fraction.

$x = - \frac{3 - 3 \sqrt{13}}{2}$

Step 1.2.8

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

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Step 1.2.8.1

Simplify the numerator.

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Step 1.2.8.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 27}}{2 \cdot 1}$

Step 1.2.8.1.2

Multiply $- 4 \cdot 1 \cdot - 27$ .

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Step 1.2.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 27}}{2 \cdot 1}$

Step 1.2.8.1.2.2

Multiply $- 4$ by $- 27$ .

$x = \frac{- 3 \pm \sqrt{9 + 108}}{2 \cdot 1}$

Step 1.2.8.1.3

Add $9$ and $108$ .

$x = \frac{- 3 \pm \sqrt{117}}{2 \cdot 1}$

Step 1.2.8.1.4

Rewrite $117$ as $3^{2} \cdot 13$ .

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Step 1.2.8.1.4.1

Factor $9$ out of $117$ .

$x = \frac{- 3 \pm \sqrt{9 (13)}}{2 \cdot 1}$

Step 1.2.8.1.4.2

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

$x = \frac{- 3 \pm \sqrt{3^{2} \cdot 13}}{2 \cdot 1}$

Step 1.2.8.1.5

Pull terms out from under the radical.

$x = \frac{- 3 \pm 3 \sqrt{13}}{2 \cdot 1}$

Step 1.2.8.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm 3 \sqrt{13}}{2}$

Step 1.2.8.3

Change the $\pm$ to $-$ .

$x = \frac{- 3 - 3 \sqrt{13}}{2}$

Step 1.2.8.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - 3 \sqrt{13}}{2}$

Step 1.2.8.5

Factor $- 1$ out of $- 3 \sqrt{13}$ .

$x = \frac{- 1 \cdot 3 - (3 \sqrt{13})}{2}$

Step 1.2.8.6

Factor $- 1$ out of $- 1 (3) - (3 \sqrt{13})$ .

$x = \frac{- 1 (3 + 3 \sqrt{13})}{2}$

Step 1.2.8.7

Move the negative in front of the fraction.

$x = - \frac{3 + 3 \sqrt{13}}{2}$

Step 1.2.9

The final answer is the combination of both solutions.

$x = - \frac{3 - 3 \sqrt{13}}{2}, - \frac{3 + 3 \sqrt{13}}{2}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{3 - 3 \sqrt{13}}{2}, 0), (- \frac{3 + 3 \sqrt{13}}{2}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 2 {(0)}^{2} + 6 (0) - 54$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 \cdot 0^{2} + 6 (0) - 54$

Step 2.2.2

Remove parentheses.

$y = 2 {(0)}^{2} + 6 (0) - 54$

Step 2.2.3

Simplify $2 {(0)}^{2} + 6 (0) - 54$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Raising $0$ to any positive power yields $0$ .

$y = 2 \cdot 0 + 6 (0) - 54$

Step 2.2.3.1.2

Multiply $2$ by $0$ .

$y = 0 + 6 (0) - 54$

Step 2.2.3.1.3

Multiply $6$ by $0$ .

$y = 0 + 0 - 54$

Step 2.2.3.2

Simplify by adding and subtracting.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 - 54$

Step 2.2.3.2.2

Subtract $54$ from $0$ .

$y = - 54$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 54)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{3 - 3 \sqrt{13}}{2}, 0), (- \frac{3 + 3 \sqrt{13}}{2}, 0)$

y-intercept(s): $(0, - 54)$

Step 4