Finite Math Examples

$f (x) = \frac{2 x^{2} + 15 x + 25}{3 x^{2} - 8 x - 16}$

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 $x$ .

$0 = \frac{2 x^{2} + 15 x + 25}{3 x^{2} - 8 x - 16}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$2 x^{2} + 15 x + 25 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Factor by grouping.

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Step 1.2.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 25 = 50$ and whose sum is $b = 15$ .

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Step 1.2.2.1.1.1

Factor $15$ out of $15 x$ .

$2 x^{2} + 15 (x) + 25 = 0$

Step 1.2.2.1.1.2

Rewrite $15$ as $5$ plus $10$

$2 x^{2} + (5 + 10) x + 25 = 0$

Step 1.2.2.1.1.3

Apply the distributive property.

$2 x^{2} + 5 x + 10 x + 25 = 0$

Step 1.2.2.1.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1.2.1

Group the first two terms and the last two terms.

$(2 x^{2} + 5 x) + 10 x + 25 = 0$

Step 1.2.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$x (2 x + 5) + 5 (2 x + 5) = 0$

Step 1.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 5$ .

$(2 x + 5) (x + 5) = 0$

Step 1.2.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x + 5 = 0$

$x + 5 = 0$

Step 1.2.2.3

Set $2 x + 5$ equal to $0$ and solve for $x$ .

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Step 1.2.2.3.1

Set $2 x + 5$ equal to $0$ .

$2 x + 5 = 0$

Step 1.2.2.3.2

Solve $2 x + 5 = 0$ for $x$ .

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Step 1.2.2.3.2.1

Subtract $5$ from both sides of the equation.

$2 x = - 5$

Step 1.2.2.3.2.2

Divide each term in $2 x = - 5$ by $2$ and simplify.

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Step 1.2.2.3.2.2.1

Divide each term in $2 x = - 5$ by $2$ .

$\frac{2 x}{2} = \frac{- 5}{2}$

Step 1.2.2.3.2.2.2

Simplify the left side.

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Step 1.2.2.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.3.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 5}{2}$

Step 1.2.2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{2}$

Step 1.2.2.3.2.2.3

Simplify the right side.

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Step 1.2.2.3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{2}$

Step 1.2.2.4

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 1.2.2.4.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 1.2.2.4.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 1.2.2.5

The final solution is all the values that make $(2 x + 5) (x + 5) = 0$ true.

$x = - \frac{5}{2}, - 5$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{5}{2}, 0), (- 5, 0)$

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 $y$ .

$y = \frac{2 {(0)}^{2} + 15 (0) + 25}{3 {(0)}^{2} - 8 \cdot 0 - 16}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{2 \cdot 0^{2} + 15 (0) + 25}{3 {(0)}^{2} - 8 \cdot 0 - 16}$

Step 2.2.2

Remove parentheses.

$y = \frac{2 \cdot 0^{2} + 15 (0) + 25}{3 \cdot 0^{2} - 8 \cdot 0 - 16}$

Step 2.2.3

Remove parentheses.

$y = \frac{2 {(0)}^{2} + 15 (0) + 25}{3 {(0)}^{2} - 8 \cdot 0 - 16}$

Step 2.2.4

Simplify $\frac{2 {(0)}^{2} + 15 (0) + 25}{3 {(0)}^{2} - 8 \cdot 0 - 16}$ .

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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 $0$ .

$y = \frac{2 \cdot 0 + 15 (0) + 25}{3 \cdot 0^{2} - 8 \cdot 0 - 16}$

Step 2.2.4.1.2

Multiply $2$ by $0$ .

$y = \frac{0 + 15 (0) + 25}{3 \cdot 0^{2} - 8 \cdot 0 - 16}$

Step 2.2.4.1.3

Multiply $15$ by $0$ .

$y = \frac{0 + 0 + 25}{3 \cdot 0^{2} - 8 \cdot 0 - 16}$

Step 2.2.4.1.4

Add $0$ and $0$ .

$y = \frac{0 + 25}{3 \cdot 0^{2} - 8 \cdot 0 - 16}$

Step 2.2.4.1.5

Add $0$ and $25$ .

$y = \frac{25}{3 \cdot 0^{2} - 8 \cdot 0 - 16}$

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 $0$ .

$y = \frac{25}{3 \cdot 0 - 8 \cdot 0 - 16}$

Step 2.2.4.2.2

Multiply $3$ by $0$ .

$y = \frac{25}{0 - 8 \cdot 0 - 16}$

Step 2.2.4.2.3

Multiply $- 8$ by $0$ .

$y = \frac{25}{0 + 0 - 16}$

Step 2.2.4.2.4

Add $0$ and $0$ .

$y = \frac{25}{0 - 16}$

Step 2.2.4.2.5

Subtract $16$ from $0$ .

$y = \frac{25}{- 16}$

Step 2.2.4.3

Move the negative in front of the fraction.

$y = - \frac{25}{16}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{25}{16})$

Step 3

List the intersections.

x-intercept(s): $(- \frac{5}{2}, 0), (- 5, 0)$

y-intercept(s): $(0, - \frac{25}{16})$

Step 4