Algebra Examples

$f (x) = \frac{x^{2} - 25}{2 x^{2} + 13 x + 15}$

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 $x$ .

$0 = \frac{x^{2} - 25}{2 x^{2} + 13 x + 15}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$x^{2} - 25 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Add $25$ to both sides of the equation.

$x^{2} = 25$

Step 1.2.2.2

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

$x = \pm \sqrt{25}$

Step 1.2.2.3

Simplify $\pm \sqrt{25}$ .

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Step 1.2.2.3.1

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

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

Step 1.2.2.3.2

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

$x = \pm 5$

Step 1.2.2.4

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

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Step 1.2.2.4.1

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

$x = 5$

Step 1.2.2.4.2

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

$x = - 5$

Step 1.2.2.4.3

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

$x = 5, - 5$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{x^{2} - 25}{2 x^{2} + 13 x + 15}$ true.

$x = 5$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

$y = \frac{{(0)}^{2} - 25}{2 {(0)}^{2} + 13 (0) + 15}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{0^{2} - 25}{2 {(0)}^{2} + 13 (0) + 15}$

Step 2.2.2

Remove parentheses.

$y = \frac{0^{2} - 25}{2 \cdot 0^{2} + 13 (0) + 15}$

Step 2.2.3

Remove parentheses.

$y = \frac{{(0)}^{2} - 25}{2 {(0)}^{2} + 13 (0) + 15}$

Step 2.2.4

Simplify $\frac{{(0)}^{2} - 25}{2 {(0)}^{2} + 13 (0) + 15}$ .

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Step 2.2.4.1

Simplify the numerator.

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Step 2.2.4.1.1

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

$y = \frac{0 - 25}{2 \cdot 0^{2} + 13 (0) + 15}$

Step 2.2.4.1.2

Subtract $25$ from $0$ .

$y = \frac{- 25}{2 \cdot 0^{2} + 13 (0) + 15}$

Step 2.2.4.2

Simplify the denominator.

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Step 2.2.4.2.1

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

$y = \frac{- 25}{2 \cdot 0 + 13 (0) + 15}$

Step 2.2.4.2.2

Multiply $2$ by $0$ .

$y = \frac{- 25}{0 + 13 (0) + 15}$

Step 2.2.4.2.3

Multiply $13$ by $0$ .

$y = \frac{- 25}{0 + 0 + 15}$

Step 2.2.4.2.4

Add $0$ and $0$ .

$y = \frac{- 25}{0 + 15}$

Step 2.2.4.2.5

Add $0$ and $15$ .

$y = \frac{- 25}{15}$

Step 2.2.4.3

Reduce the expression by cancelling the common factors.

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Step 2.2.4.3.1

Cancel the common factor of $- 25$ and $15$ .

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Step 2.2.4.3.1.1

Factor $5$ out of $- 25$ .

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

Step 2.2.4.3.1.2

Cancel the common factors.

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Step 2.2.4.3.1.2.1

Factor $5$ out of $15$ .

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

Step 2.2.4.3.1.2.2

Cancel the common factor.

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

Step 2.2.4.3.1.2.3

Rewrite the expression.

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

Step 2.2.4.3.2

Move the negative in front of the fraction.

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

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{5}{3})$

Step 3

List the intersections.

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

y-intercept(s): $(0, - \frac{5}{3})$

Step 4