Trigonometry Examples

$y = - 3 x^{2} + 6 x + 5$

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 $x$ .

$0 = - 3 x^{2} + 6 x + 5$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 3 x^{2} + 6 x + 5 = 0$ .

$- 3 x^{2} + 6 x + 5 = 0$

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 $a = - 3$ , $b = 6$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (- 3 \cdot 5)}}{2 \cdot - 3}$

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 $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 3 \cdot 5}}{2 \cdot - 3}$

Step 1.2.4.1.2

Multiply $- 4 \cdot - 3 \cdot 5$ .

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Step 1.2.4.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 6 \pm \sqrt{36 + 12 \cdot 5}}{2 \cdot - 3}$

Step 1.2.4.1.2.2

Multiply $12$ by $5$ .

$x = \frac{- 6 \pm \sqrt{36 + 60}}{2 \cdot - 3}$

Step 1.2.4.1.3

Add $36$ and $60$ .

$x = \frac{- 6 \pm \sqrt{96}}{2 \cdot - 3}$

Step 1.2.4.1.4

Rewrite $96$ as $4^{2} \cdot 6$ .

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Step 1.2.4.1.4.1

Factor $16$ out of $96$ .

$x = \frac{- 6 \pm \sqrt{16 (6)}}{2 \cdot - 3}$

Step 1.2.4.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 6 \pm \sqrt{4^{2} \cdot 6}}{2 \cdot - 3}$

Step 1.2.4.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 4 \sqrt{6}}{2 \cdot - 3}$

Step 1.2.4.2

Multiply $2$ by $- 3$ .

$x = \frac{- 6 \pm 4 \sqrt{6}}{- 6}$

Step 1.2.4.3

Simplify $\frac{- 6 \pm 4 \sqrt{6}}{- 6}$ .

$x = \frac{3 \pm 2 \sqrt{6}}{3}$

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 $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 3 \cdot 5}}{2 \cdot - 3}$

Step 1.2.5.1.2

Multiply $- 4 \cdot - 3 \cdot 5$ .

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Step 1.2.5.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 6 \pm \sqrt{36 + 12 \cdot 5}}{2 \cdot - 3}$

Step 1.2.5.1.2.2

Multiply $12$ by $5$ .

$x = \frac{- 6 \pm \sqrt{36 + 60}}{2 \cdot - 3}$

Step 1.2.5.1.3

Add $36$ and $60$ .

$x = \frac{- 6 \pm \sqrt{96}}{2 \cdot - 3}$

Step 1.2.5.1.4

Rewrite $96$ as $4^{2} \cdot 6$ .

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Step 1.2.5.1.4.1

Factor $16$ out of $96$ .

$x = \frac{- 6 \pm \sqrt{16 (6)}}{2 \cdot - 3}$

Step 1.2.5.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 6 \pm \sqrt{4^{2} \cdot 6}}{2 \cdot - 3}$

Step 1.2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 4 \sqrt{6}}{2 \cdot - 3}$

Step 1.2.5.2

Multiply $2$ by $- 3$ .

$x = \frac{- 6 \pm 4 \sqrt{6}}{- 6}$

Step 1.2.5.3

Simplify $\frac{- 6 \pm 4 \sqrt{6}}{- 6}$ .

$x = \frac{3 \pm 2 \sqrt{6}}{3}$

Step 1.2.5.4

Change the $\pm$ to $+$ .

$x = \frac{3 + 2 \sqrt{6}}{3}$

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 $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 3 \cdot 5}}{2 \cdot - 3}$

Step 1.2.6.1.2

Multiply $- 4 \cdot - 3 \cdot 5$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 6 \pm \sqrt{36 + 12 \cdot 5}}{2 \cdot - 3}$

Step 1.2.6.1.2.2

Multiply $12$ by $5$ .

$x = \frac{- 6 \pm \sqrt{36 + 60}}{2 \cdot - 3}$

Step 1.2.6.1.3

Add $36$ and $60$ .

$x = \frac{- 6 \pm \sqrt{96}}{2 \cdot - 3}$

Step 1.2.6.1.4

Rewrite $96$ as $4^{2} \cdot 6$ .

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Step 1.2.6.1.4.1

Factor $16$ out of $96$ .

$x = \frac{- 6 \pm \sqrt{16 (6)}}{2 \cdot - 3}$

Step 1.2.6.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 6 \pm \sqrt{4^{2} \cdot 6}}{2 \cdot - 3}$

Step 1.2.6.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 4 \sqrt{6}}{2 \cdot - 3}$

Step 1.2.6.2

Multiply $2$ by $- 3$ .

$x = \frac{- 6 \pm 4 \sqrt{6}}{- 6}$

Step 1.2.6.3

Simplify $\frac{- 6 \pm 4 \sqrt{6}}{- 6}$ .

$x = \frac{3 \pm 2 \sqrt{6}}{3}$

Step 1.2.6.4

Change the $\pm$ to $-$ .

$x = \frac{3 - 2 \sqrt{6}}{3}$

Step 1.2.7

The final answer is the combination of both solutions.

$x = \frac{3 + 2 \sqrt{6}}{3}, \frac{3 - 2 \sqrt{6}}{3}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{3 + 2 \sqrt{6}}{3}, 0), (\frac{3 - 2 \sqrt{6}}{3}, 0)$

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 $y$ .

$y = - 3 {(0)}^{2} + 6 (0) + 5$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 3 \cdot 0^{2} + 6 (0) + 5$

Step 2.2.2

Remove parentheses.

$y = - 3 {(0)}^{2} + 6 (0) + 5$

Step 2.2.3

Simplify $- 3 {(0)}^{2} + 6 (0) + 5$ .

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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 $0$ .

$y = - 3 \cdot 0 + 6 (0) + 5$

Step 2.2.3.1.2

Multiply $- 3$ by $0$ .

$y = 0 + 6 (0) + 5$

Step 2.2.3.1.3

Multiply $6$ by $0$ .

$y = 0 + 0 + 5$

Step 2.2.3.2

Simplify by adding numbers.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 + 5$

Step 2.2.3.2.2

Add $0$ and $5$ .

$y = 5$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 5)$

Step 3

List the intersections.

x-intercept(s): $(\frac{3 + 2 \sqrt{6}}{3}, 0), (\frac{3 - 2 \sqrt{6}}{3}, 0)$

y-intercept(s): $(0, 5)$

Step 4