Find the x and y Intercepts f(x)=(-3x^2+9x-6)/(2x^2-8)

$f (x) = \frac{- 3 x^{2} + 9 x - 6}{2 x^{2} - 8}$

Step 1

Find the x-intercepts.

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

$0 = \frac{- 3 x^{2} + 9 x - 6}{2 x^{2} - 8}$

Solve the equation.

Set the numerator equal to zero.

$- 3 x^{2} + 9 x - 6 = 0$

Solve the equation for $x$ .

Factor the left side of the equation.

Factor $- 3$ out of $- 3 x^{2} + 9 x - 6$ .

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

$- 3 x^{2} + 9 x - 6 = 0$

Factor $- 3$ out of $9 x$ .

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

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

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

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

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

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

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

Factor.

Factor $x^{2} - 3 x + 2$ 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 $2$ and whose sum is $- 3$ .

$- 2, - 1$

Write the factored form using these integers.

$- 3 ((x - 2) (x - 1)) = 0$

Remove unnecessary parentheses.

$- 3 (x - 2) (x - 1) = 0$

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

$x - 2 = 0$

$x - 1 = 0$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $- 3 (x - 2) (x - 1) = 0$ true.

$x = 2, 1$

Exclude the solutions that do not make $0 = \frac{- 3 x^{2} + 9 x - 6}{2 x^{2} - 8}$ true.

$x = 1$

x-intercept(s) in point form.

x-intercept(s): $(1, 0)$

Step 2

Find the y-intercepts.

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

$y = \frac{- 3 {(0)}^{2} + 9 (0) - 6}{2 {(0)}^{2} - 8}$

Solve the equation.

Remove parentheses.

$y = \frac{- 3 \cdot 0^{2} + 9 (0) - 6}{2 {(0)}^{2} - 8}$

Remove parentheses.

$y = \frac{- 3 \cdot 0^{2} + 9 (0) - 6}{2 \cdot 0^{2} - 8}$

Remove parentheses.

$y = \frac{- 3 {(0)}^{2} + 9 (0) - 6}{2 {(0)}^{2} - 8}$

Simplify $\frac{- 3 {(0)}^{2} + 9 (0) - 6}{2 {(0)}^{2} - 8}$ .

Simplify the numerator.

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

$y = \frac{- 3 \cdot 0 + 9 (0) - 6}{2 \cdot 0^{2} - 8}$

Multiply $- 3$ by $0$ .

$y = \frac{0 + 9 (0) - 6}{2 \cdot 0^{2} - 8}$

Multiply $9$ by $0$ .

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

Add $0$ and $0$ .

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

Subtract $6$ from $0$ .

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

Simplify the denominator.

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

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

Multiply $2$ by $0$ .

$y = \frac{- 6}{0 - 8}$

Subtract $8$ from $0$ .

$y = \frac{- 6}{- 8}$

Cancel the common factor of $- 6$ and $- 8$ .

Factor $- 2$ out of $- 6$ .

$y = \frac{- 2 (3)}{- 8}$

Cancel the common factors.

Factor $- 2$ out of $- 8$ .

$y = \frac{- 2 \cdot 3}{- 2 \cdot 4}$

Cancel the common factor.

$y = \frac{- 2 \cdot 3}{- 2 \cdot 4}$

Rewrite the expression.

$y = \frac{3}{4}$

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{3}{4})$

Step 3

List the intersections.

x-intercept(s): $(1, 0)$

y-intercept(s): $(0, \frac{3}{4})$

Step 4

Find the x and y Intercepts f(x)=(-3x^2+9x-6)/(2x^2-8)

Related Questions

Support LCMGCF.com by disabling your adblocker.