Find the x and y Intercepts f(x)=(2x^2+17x+30)/(2x^2-18)

$f (x) = \frac{2 x^{2} + 17 x + 30}{2 x^{2} - 18}$

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{2 x^{2} + 17 x + 30}{2 x^{2} - 18}$

Solve the equation.

Set the numerator equal to zero.

$2 x^{2} + 17 x + 30 = 0$

Solve the equation for $x$ .

Factor by grouping.

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 $a \cdot c = 2 \cdot 30 = 60$ and whose sum is $b = 17$ .

Factor $17$ out of $17 x$ .

$2 x^{2} + 17 (x) + 30 = 0$

Rewrite $17$ as $5$ plus $12$

$2 x^{2} + (5 + 12) x + 30 = 0$

Apply the distributive property.

$2 x^{2} + 5 x + 12 x + 30 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 x^{2} + 5 x) + 12 x + 30 = 0$

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

$x (2 x + 5) + 6 (2 x + 5) = 0$

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

$(2 x + 5) (x + 6) = 0$

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

$2 x + 5 = 0$

$x + 6 = 0$

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

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

$2 x + 5 = 0$

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

Subtract $5$ from both sides of the equation.

$2 x = - 5$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $(2 x + 5) (x + 6) = 0$ true.

$x = - \frac{5}{2}, - 6$

x-intercept(s) in point form.

x-intercept(s): $(- \frac{5}{2}, 0), (- 6, 0)$

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{2 {(0)}^{2} + 17 (0) + 30}{2 {(0)}^{2} - 18}$

Solve the equation.

Remove parentheses.

$y = \frac{2 \cdot 0^{2} + 17 (0) + 30}{2 {(0)}^{2} - 18}$

Remove parentheses.

$y = \frac{2 \cdot 0^{2} + 17 (0) + 30}{2 \cdot 0^{2} - 18}$

Remove parentheses.

$y = \frac{2 {(0)}^{2} + 17 (0) + 30}{2 {(0)}^{2} - 18}$

Simplify $\frac{2 {(0)}^{2} + 17 (0) + 30}{2 {(0)}^{2} - 18}$ .

Cancel the common factor of $2 {(0)}^{2} + 17 (0) + 30$ and $2 {(0)}^{2} - 18$ .

Factor $2$ out of $2 {(0)}^{2}$ .

$y = \frac{2 ({(0)}^{2}) + 17 (0) + 30}{2 {(0)}^{2} - 18}$

Factor $2$ out of $17 (0)$ .

$y = \frac{2 ({(0)}^{2}) + 2 (17 \cdot (0)) + 30}{2 {(0)}^{2} - 18}$

Factor $2$ out of $2 ({(0)}^{2}) + 2 (17 \cdot (0))$ .

$y = \frac{2 ({(0)}^{2} + 17 \cdot (0)) + 30}{2 {(0)}^{2} - 18}$

Factor $2$ out of $30$ .

$y = \frac{2 ({(0)}^{2} + 17 \cdot (0)) + 2 \cdot 15}{2 {(0)}^{2} - 18}$

Factor $2$ out of $2 ({(0)}^{2} + 17 \cdot (0)) + 2 (15)$ .

$y = \frac{2 ({(0)}^{2} + 17 \cdot (0) + 15)}{2 {(0)}^{2} - 18}$

Cancel the common factors.

Factor $2$ out of $2 {(0)}^{2}$ .

$y = \frac{2 ({(0)}^{2} + 17 \cdot (0) + 15)}{2 ({(0)}^{2}) - 18}$

Factor $2$ out of $- 18$ .

$y = \frac{2 ({(0)}^{2} + 17 \cdot (0) + 15)}{2 {(0)}^{2} + 2 \cdot - 9}$

Factor $2$ out of $2 {(0)}^{2} + 2 \cdot - 9$ .

$y = \frac{2 ({(0)}^{2} + 17 \cdot (0) + 15)}{2 ({(0)}^{2} - 9)}$

Cancel the common factor.

$y = \frac{2 ({(0)}^{2} + 17 \cdot (0) + 15)}{2 ({(0)}^{2} - 9)}$

Rewrite the expression.

$y = \frac{{(0)}^{2} + 17 \cdot (0) + 15}{{(0)}^{2} - 9}$

Simplify the numerator.

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

$y = \frac{0 + 17 \cdot (0) + 15}{0^{2} - 9}$

Multiply $17$ by $0$ .

$y = \frac{0 + 0 + 15}{0^{2} - 9}$

Add $0$ and $0$ .

$y = \frac{0 + 15}{0^{2} - 9}$

Add $0$ and $15$ .

$y = \frac{15}{0^{2} - 9}$

Simplify the denominator.

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

$y = \frac{15}{0 - 9}$

Subtract $9$ from $0$ .

$y = \frac{15}{- 9}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $15$ and $- 9$ .

Factor $3$ out of $15$ .

$y = \frac{3 (5)}{- 9}$

Cancel the common factors.

Factor $3$ out of $- 9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

y-intercept(s) in point form.

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

List the intersections.

x-intercept(s): $(- \frac{5}{2}, 0), (- 6, 0)$

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

Find the x and y Intercepts f(x)=(2x^2+17x+30)/(2x^2-18)

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