Find the x and y Intercepts f(x)=(2x^2+5x-12)/(2x+8)

$f (x) = \frac{2 x^{2} + 5 x - 12}{2 x + 8}$

Find the x-intercepts.

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

$0 = \frac{2 x^{2} + 5 x - 12}{2 x + 8}$

Solve the equation.

Set the numerator equal to zero.

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

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 $a \cdot c = 2 \cdot - 12 = - 24$ and whose sum is $b = 5$ .

Factor $5$ out of $5 x$ .

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

Rewrite $5$ as $- 3$ plus $8$

$2 x^{2} + (- 3 + 8) x - 12 = 0$

Apply the distributive property.

$2 x^{2} - 3 x + 8 x - 12 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 x^{2} - 3 x) + 8 x - 12 = 0$

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

$x (2 x - 3) + 4 (2 x - 3) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x - 3$ .

$(2 x - 3) (x + 4) = 0$

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

$2 x - 3 = 0$

$x + 4 = 0$

Set $2 x - 3$ equal to $0$ and solve for $x$ .

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Solve $2 x - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$2 x = 3$

Divide each term in $2 x = 3$ by $2$ and simplify.

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $(2 x - 3) (x + 4) = 0$ true.

$x = \frac{3}{2}, - 4$

Exclude the solutions that do not make $0 = \frac{2 x^{2} + 5 x - 12}{2 x + 8}$ true.

$x = \frac{3}{2}$

x-intercept(s) in point form.

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

Find the y-intercepts.

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

$y = \frac{2 {(0)}^{2} + 5 (0) - 12}{2 (0) + 8}$

Solve the equation.

Remove parentheses.

$y = \frac{2 \cdot 0^{2} + 5 (0) - 12}{2 (0) + 8}$

Remove parentheses.

$y = \frac{2 {(0)}^{2} + 5 (0) - 12}{2 (0) + 8}$

Simplify $\frac{2 {(0)}^{2} + 5 (0) - 12}{2 (0) + 8}$ .

Cancel the common factor of $2 {(0)}^{2} + 5 (0) - 12$ and $2 (0) + 8$ .

Factor $- 1$ out of $2 (0)$ .

$y = \frac{2 {(0)}^{2} + 5 (0) - 12}{- (- 2 \cdot 0) + 8}$

Rewrite $8$ as $- 1 (- 8)$ .

$y = \frac{2 {(0)}^{2} + 5 (0) - 12}{- (- 2 \cdot 0) - 1 (- 8)}$

Factor $- 1$ out of $- (- 2 \cdot 0) - 1 (- 8)$ .

$y = \frac{2 {(0)}^{2} + 5 (0) - 12}{- (- 2 \cdot 0 - 8)}$

Rewrite $- (- 2 \cdot 0 - 8)$ as $- 1 (- 2 \cdot 0 - 8)$ .

$y = \frac{2 {(0)}^{2} + 5 (0) - 12}{- 1 (- 2 \cdot 0 - 8)}$

Reorder terms.

$y = \frac{2 \cdot 0^{2} + 5 (0) - 12}{- 1 (- 8 - 2 \cdot 0)}$

Factor $2$ out of $2 \cdot 0^{2}$ .

$y = \frac{2 (0^{2}) + 5 (0) - 12}{- 1 (- 8 - 2 \cdot 0)}$

Factor $2$ out of $5 (0)$ .

$y = \frac{2 (0^{2}) + 2 (5 \cdot (0)) - 12}{- 1 (- 8 - 2 \cdot 0)}$

Factor $2$ out of $2 (0^{2}) + 2 (5 \cdot (0))$ .

$y = \frac{2 (0^{2} + 5 \cdot (0)) - 12}{- 1 (- 8 - 2 \cdot 0)}$

Factor $2$ out of $- 12$ .

$y = \frac{2 (0^{2} + 5 \cdot (0)) + 2 (- 6)}{- 1 (- 8 - 2 \cdot 0)}$

Factor $2$ out of $2 (0^{2} + 5 \cdot (0)) + 2 (- 6)$ .

$y = \frac{2 (0^{2} + 5 \cdot (0) - 6)}{- 1 (- 8 - 2 \cdot 0)}$

Cancel the common factors.

Factor $2$ out of $- 1 (- 8 - 2 \cdot 0)$ .

$y = \frac{2 (0^{2} + 5 \cdot (0) - 6)}{2 (- 1 (- 4 - 0))}$

Cancel the common factor.

$y = \frac{2 (0^{2} + 5 \cdot (0) - 6)}{2 (- 1 (- 4 - 0))}$

Rewrite the expression.

$y = \frac{0^{2} + 5 \cdot (0) - 6}{- 1 (- 4 - 0)}$

Simplify the numerator.

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

$y = \frac{0 + 5 \cdot (0) - 6}{- 1 (- 4 - 0)}$

Multiply $5$ by $0$ .

$y = \frac{0 + 0 - 6}{- 1 (- 4 - 0)}$

Add $0$ and $0$ .

$y = \frac{0 - 6}{- 1 (- 4 - 0)}$

Subtract $6$ from $0$ .

$y = \frac{- 6}{- 1 (- 4 - 0)}$

Simplify the denominator.

Multiply $- 1$ by $0$ .

$y = \frac{- 6}{- 1 (- 4 + 0)}$

Add $- 4$ and $0$ .

$y = \frac{- 6}{- 1 \cdot - 4}$

Reduce the expression by cancelling the common factors.

Multiply $- 1$ by $- 4$ .

$y = \frac{- 6}{4}$

Cancel the common factor of $- 6$ and $4$ .

Factor $2$ out of $- 6$ .

$y = \frac{2 (- 3)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = \frac{2 \cdot - 3}{2 \cdot 2}$

Cancel the common factor.

$y = \frac{2 \cdot - 3}{2 \cdot 2}$

Rewrite the expression.

$y = \frac{- 3}{2}$

Move the negative in front of the fraction.

$y = - \frac{3}{2}$

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{3}{2})$

List the intersections.

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

y-intercept(s): $(0, - \frac{3}{2})$

Find the x and y Intercepts f(x)=(2x^2+5x-12)/(2x+8)

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