Find the x and y Intercepts (3x^2-6x-24)/(2x^2-8)

$\frac{3 x^{2} - 6 x - 24}{2 x^{2} - 8}$

Step 1

Write $\frac{3 x^{2} - 6 x - 24}{2 x^{2} - 8}$ as an equation.

$y = \frac{3 x^{2} - 6 x - 24}{2 x^{2} - 8}$

Step 2

Find the x-intercepts.

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

$0 = \frac{3 x^{2} - 6 x - 24}{2 x^{2} - 8}$

Solve the equation.

Set the numerator equal to zero.

$3 x^{2} - 6 x - 24 = 0$

Solve the equation for $x$ .

Factor the left side of the equation.

Factor $3$ out of $3 x^{2} - 6 x - 24$ .

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) - 6 x - 24 = 0$

Factor $3$ out of $- 6 x$ .

$3 (x^{2}) + 3 (- 2 x) - 24 = 0$

Factor $3$ out of $- 24$ .

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

Factor $3$ out of $3 x^{2} + 3 (- 2 x)$ .

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

Factor $3$ out of $3 (x^{2} - 2 x) + 3 \cdot - 8$ .

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

Factor.

Factor $x^{2} - 2 x - 8$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

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

$x - 4 = 0$

$x + 2 = 0$

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

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

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

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

$x = 4, - 2$

Exclude the solutions that do not make $0 = \frac{3 x^{2} - 6 x - 24}{2 x^{2} - 8}$ true.

$x = 4$

x-intercept(s) in point form.

x-intercept(s): $(4, 0)$

Step 3

Find the y-intercepts.

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

$y = \frac{3 {(0)}^{2} - 6 \cdot 0 - 24}{2 {(0)}^{2} - 8}$

Solve the equation.

Remove parentheses.

$y = \frac{3 \cdot 0^{2} - 6 \cdot 0 - 24}{2 {(0)}^{2} - 8}$

Remove parentheses.

$y = \frac{3 \cdot 0^{2} - 6 \cdot 0 - 24}{2 \cdot 0^{2} - 8}$

Remove parentheses.

$y = \frac{3 {(0)}^{2} - 6 \cdot 0 - 24}{2 {(0)}^{2} - 8}$

Simplify $\frac{3 {(0)}^{2} - 6 \cdot 0 - 24}{2 {(0)}^{2} - 8}$ .

Simplify the numerator.

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

$y = \frac{3 \cdot 0 - 6 \cdot 0 - 24}{2 \cdot 0^{2} - 8}$

Multiply $3$ by $0$ .

$y = \frac{0 - 6 \cdot 0 - 24}{2 \cdot 0^{2} - 8}$

Multiply $- 6$ by $0$ .

$y = \frac{0 + 0 - 24}{2 \cdot 0^{2} - 8}$

Add $0$ and $0$ .

$y = \frac{0 - 24}{2 \cdot 0^{2} - 8}$

Subtract $24$ from $0$ .

$y = \frac{- 24}{2 \cdot 0^{2} - 8}$

Simplify the denominator.

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

$y = \frac{- 24}{2 \cdot 0 - 8}$

Multiply $2$ by $0$ .

$y = \frac{- 24}{0 - 8}$

Subtract $8$ from $0$ .

$y = \frac{- 24}{- 8}$

Divide $- 24$ by $- 8$ .

$y = 3$

y-intercept(s) in point form.

y-intercept(s): $(0, 3)$

Step 4

List the intersections.

x-intercept(s): $(4, 0)$

y-intercept(s): $(0, 3)$

Step 5

Find the x and y Intercepts (3x^2-6x-24)/(2x^2-8)

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