Trigonometry Examples

$y = \frac{4 x^{2} - 100}{2 x^{2} + x - 15}$

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 $x$ .

$0 = \frac{4 x^{2} - 100}{2 x^{2} + x - 15}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$4 x^{2} - 100 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Add $100$ to both sides of the equation.

$4 x^{2} = 100$

Step 1.2.2.2

Divide each term in $4 x^{2} = 100$ by $4$ and simplify.

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Step 1.2.2.2.1

Divide each term in $4 x^{2} = 100$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{100}{4}$

Step 1.2.2.2.2

Simplify the left side.

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Step 1.2.2.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.2.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{100}{4}$

Step 1.2.2.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{100}{4}$

Step 1.2.2.2.3

Simplify the right side.

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Step 1.2.2.2.3.1

Divide $100$ by $4$ .

$x^{2} = 25$

Step 1.2.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{25}$

Step 1.2.2.4

Simplify $\pm \sqrt{25}$ .

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Step 1.2.2.4.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2}}$

Step 1.2.2.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 5$

Step 1.2.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 5$

Step 1.2.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 5$

Step 1.2.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 5, - 5$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

$y = \frac{4 {(0)}^{2} - 100}{2 {(0)}^{2} + 0 - 15}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{4 {(0)}^{2} - 100}{2 {(0)}^{2} + 0 - 15}$

Step 2.2.2

Remove parentheses.

$y = \frac{4 \cdot 0^{2} - 100}{2 {(0)}^{2} + 0 - 15}$

Step 2.2.3

Remove parentheses.

$y = \frac{4 \cdot 0^{2} - 100}{2 \cdot 0^{2} + 0 - 15}$

Step 2.2.4

Remove parentheses.

$y = \frac{4 {(0)}^{2} - 100}{2 {(0)}^{2} + 0 - 15}$

Step 2.2.5

Simplify $\frac{4 {(0)}^{2} - 100}{2 {(0)}^{2} + 0 - 15}$ .

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Step 2.2.5.1

Simplify the numerator.

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Step 2.2.5.1.1

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

$y = \frac{4 \cdot 0 - 100}{2 \cdot 0^{2} + 0 - 15}$

Step 2.2.5.1.2

Multiply $4$ by $0$ .

$y = \frac{0 - 100}{2 \cdot 0^{2} + 0 - 15}$

Step 2.2.5.1.3

Subtract $100$ from $0$ .

$y = \frac{- 100}{2 \cdot 0^{2} + 0 - 15}$

Step 2.2.5.2

Simplify the denominator.

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Step 2.2.5.2.1

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

$y = \frac{- 100}{2 \cdot 0 + 0 - 15}$

Step 2.2.5.2.2

Multiply $2$ by $0$ .

$y = \frac{- 100}{0 + 0 - 15}$

Step 2.2.5.2.3

Add $0$ and $0$ .

$y = \frac{- 100}{0 - 15}$

Step 2.2.5.2.4

Subtract $15$ from $0$ .

$y = \frac{- 100}{- 15}$

Step 2.2.5.3

Cancel the common factor of $- 100$ and $- 15$ .

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Step 2.2.5.3.1

Factor $- 5$ out of $- 100$ .

$y = \frac{- 5 (20)}{- 15}$

Step 2.2.5.3.2

Cancel the common factors.

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Step 2.2.5.3.2.1

Factor $- 5$ out of $- 15$ .

$y = \frac{- 5 \cdot 20}{- 5 \cdot 3}$

Step 2.2.5.3.2.2

Cancel the common factor.

$y = \frac{- 5 \cdot 20}{- 5 \cdot 3}$

Step 2.2.5.3.2.3

Rewrite the expression.

$y = \frac{20}{3}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{20}{3})$

Step 3

List the intersections.

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

y-intercept(s): $(0, \frac{20}{3})$

Step 4