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

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

Find the x-intercepts.

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

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

Solve the equation.

Set the numerator equal to zero.

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

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 $a \cdot c = 2 \cdot 9 = 18$ and whose sum is $b = - 9$ .

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

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

Rewrite $- 9$ as $- 3$ plus $- 6$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

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

$2 x - 3 = 0$

$x - 3 = 0$

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

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

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

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

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

Solve the equation.

Remove parentheses.

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

Remove parentheses.

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

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

Simplify the numerator.

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

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

Multiply $2$ by $0$ .

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

Multiply $- 9$ by $0$ .

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

Add $0$ and $0$ .

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

Add $0$ and $9$ .

$y = \frac{9}{2 (0) - 6}$

Simplify the denominator.

Multiply $2$ by $0$ .

$y = \frac{9}{0 - 6}$

Subtract $6$ from $0$ .

$y = \frac{9}{- 6}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $9$ and $- 6$ .

Factor $3$ out of $9$ .

$y = \frac{3 (3)}{- 6}$

Cancel the common factors.

Factor $3$ out of $- 6$ .

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

Cancel the common factor.

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

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^2-9x+9)/(2x-6)

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