Trigonometry Examples

$f (x) = 2 x^{2} + 3 x - 4$

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 $x$ .

$0 = 2 x^{2} + 3 x - 4$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 x^{2} + 3 x - 4 = 0$ .

$2 x^{2} + 3 x - 4 = 0$

Step 1.2.2

Use the quadratic formula to find the solutions.

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

Step 1.2.3

Substitute the values $a = 2$ , $b = 3$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

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

Step 1.2.4

Simplify.

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Step 1.2.4.1

Simplify the numerator.

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Step 1.2.4.1.1

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

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

Step 1.2.4.1.2

Multiply $- 4 \cdot 2 \cdot - 4$ .

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Step 1.2.4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 8 \cdot - 4}}{2 \cdot 2}$

Step 1.2.4.1.2.2

Multiply $- 8$ by $- 4$ .

$x = \frac{- 3 \pm \sqrt{9 + 32}}{2 \cdot 2}$

Step 1.2.4.1.3

Add $9$ and $32$ .

$x = \frac{- 3 \pm \sqrt{41}}{2 \cdot 2}$

Step 1.2.4.2

Multiply $2$ by $2$ .

$x = \frac{- 3 \pm \sqrt{41}}{4}$

Step 1.2.5

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

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Step 1.2.5.1

Simplify the numerator.

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Step 1.2.5.1.1

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

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

Step 1.2.5.1.2

Multiply $- 4 \cdot 2 \cdot - 4$ .

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Step 1.2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 8 \cdot - 4}}{2 \cdot 2}$

Step 1.2.5.1.2.2

Multiply $- 8$ by $- 4$ .

$x = \frac{- 3 \pm \sqrt{9 + 32}}{2 \cdot 2}$

Step 1.2.5.1.3

Add $9$ and $32$ .

$x = \frac{- 3 \pm \sqrt{41}}{2 \cdot 2}$

Step 1.2.5.2

Multiply $2$ by $2$ .

$x = \frac{- 3 \pm \sqrt{41}}{4}$

Step 1.2.5.3

Change the $\pm$ to $+$ .

$x = \frac{- 3 + \sqrt{41}}{4}$

Step 1.2.5.4

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

$x = \frac{- 1 \cdot 3 + \sqrt{41}}{4}$

Step 1.2.5.5

Factor $- 1$ out of $\sqrt{41}$ .

$x = \frac{- 1 \cdot 3 - 1 (- \sqrt{41})}{4}$

Step 1.2.5.6

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{41})$ .

$x = \frac{- 1 (3 - \sqrt{41})}{4}$

Step 1.2.5.7

Move the negative in front of the fraction.

$x = - \frac{3 - \sqrt{41}}{4}$

Step 1.2.6

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

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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 $2$ .

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

Step 1.2.6.1.2

Multiply $- 4 \cdot 2 \cdot - 4$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 8 \cdot - 4}}{2 \cdot 2}$

Step 1.2.6.1.2.2

Multiply $- 8$ by $- 4$ .

$x = \frac{- 3 \pm \sqrt{9 + 32}}{2 \cdot 2}$

Step 1.2.6.1.3

Add $9$ and $32$ .

$x = \frac{- 3 \pm \sqrt{41}}{2 \cdot 2}$

Step 1.2.6.2

Multiply $2$ by $2$ .

$x = \frac{- 3 \pm \sqrt{41}}{4}$

Step 1.2.6.3

Change the $\pm$ to $-$ .

$x = \frac{- 3 - \sqrt{41}}{4}$

Step 1.2.6.4

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

$x = \frac{- 1 \cdot 3 - \sqrt{41}}{4}$

Step 1.2.6.5

Factor $- 1$ out of $- \sqrt{41}$ .

$x = \frac{- 1 \cdot 3 - (\sqrt{41})}{4}$

Step 1.2.6.6

Factor $- 1$ out of $- 1 (3) - (\sqrt{41})$ .

$x = \frac{- 1 (3 + \sqrt{41})}{4}$

Step 1.2.6.7

Move the negative in front of the fraction.

$x = - \frac{3 + \sqrt{41}}{4}$

Step 1.2.7

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{41}}{4}, - \frac{3 + \sqrt{41}}{4}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{3 - \sqrt{41}}{4}, 0), (- \frac{3 + \sqrt{41}}{4}, 0)$

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 $y$ .

$y = 2 {(0)}^{2} + 3 (0) - 4$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 \cdot 0^{2} + 3 (0) - 4$

Step 2.2.2

Remove parentheses.

$y = 2 {(0)}^{2} + 3 (0) - 4$

Step 2.2.3

Simplify $2 {(0)}^{2} + 3 (0) - 4$ .

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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 $0$ .

$y = 2 \cdot 0 + 3 (0) - 4$

Step 2.2.3.1.2

Multiply $2$ by $0$ .

$y = 0 + 3 (0) - 4$

Step 2.2.3.1.3

Multiply $3$ by $0$ .

$y = 0 + 0 - 4$

Step 2.2.3.2

Simplify by adding and subtracting.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 - 4$

Step 2.2.3.2.2

Subtract $4$ from $0$ .

$y = - 4$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 4)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{3 - \sqrt{41}}{4}, 0), (- \frac{3 + \sqrt{41}}{4}, 0)$

y-intercept(s): $(0, - 4)$

Step 4