Find the x and y Intercepts f(x)=(3x^2-9x+6)/(5x-10)

$f (x) = \frac{3 x^{2} - 9 x + 6}{5 x - 10}$

Step 1

Find the x-intercepts.

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

$0 = \frac{3 x^{2} - 9 x + 6}{5 x - 10}$

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 $3 (x^{2} - 3 x) + 3 \cdot 2$ .

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

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 $0 = \frac{3 x^{2} - 9 x + 6}{5 x - 10}$ 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 $y$ .

$y = \frac{3 {(0)}^{2} - 9 \cdot 0 + 6}{5 (0) - 10}$

Solve the equation.

Remove parentheses.

$y = \frac{3 \cdot 0^{2} - 9 \cdot 0 + 6}{5 (0) - 10}$

Remove parentheses.

$y = \frac{3 {(0)}^{2} - 9 \cdot 0 + 6}{5 (0) - 10}$

Simplify $\frac{3 {(0)}^{2} - 9 \cdot 0 + 6}{5 (0) - 10}$ .

Simplify the numerator.

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

$y = \frac{3 \cdot 0 - 9 \cdot 0 + 6}{5 (0) - 10}$

Multiply $3$ by $0$ .

$y = \frac{0 - 9 \cdot 0 + 6}{5 (0) - 10}$

Multiply $- 9$ by $0$ .

$y = \frac{0 + 0 + 6}{5 (0) - 10}$

Add $0$ and $0$ .

$y = \frac{0 + 6}{5 (0) - 10}$

Add $0$ and $6$ .

$y = \frac{6}{5 (0) - 10}$

Simplify the denominator.

Multiply $5$ by $0$ .

$y = \frac{6}{0 - 10}$

Subtract $10$ from $0$ .

$y = \frac{6}{- 10}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $6$ and $- 10$ .

Factor $2$ out of $6$ .

$y = \frac{2 (3)}{- 10}$

Cancel the common factors.

Factor $2$ out of $- 10$ .

$y = \frac{2 \cdot 3}{2 \cdot - 5}$

Cancel the common factor.

$y = \frac{2 \cdot 3}{2 \cdot - 5}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4

Find the x and y Intercepts f(x)=(3x^2-9x+6)/(5x-10)

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