Find the x and y Intercepts f(x)=(x^2-7x+12)/(-5x+15)

$f (x) = \frac{x^{2} - 7 x + 12}{- 5 x + 15}$

Find the x-intercepts.

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

$0 = \frac{x^{2} - 7 x + 12}{- 5 x + 15}$

Solve the equation.

Set the numerator equal to zero.

$x^{2} - 7 x + 12 = 0$

Solve the equation for $x$ .

Factor $x^{2} - 7 x + 12$ 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 $12$ and whose sum is $- 7$ .

$- 4, - 3$

Write the factored form using these integers.

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

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

$x - 4 = 0$

$x - 3 = 0$

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 $x - 3$ 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 $(x - 4) (x - 3) = 0$ true.

$x = 4, 3$

Exclude the solutions that do not make $0 = \frac{x^{2} - 7 x + 12}{- 5 x + 15}$ true.

$x = 4$

x-intercept(s) in point form.

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

Find the y-intercepts.

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

$y = \frac{{(0)}^{2} - 7 \cdot 0 + 12}{- 5 \cdot 0 + 15}$

Solve the equation.

Remove parentheses.

$y = \frac{0^{2} - 7 \cdot 0 + 12}{- 5 \cdot 0 + 15}$

Remove parentheses.

$y = \frac{{(0)}^{2} - 7 \cdot 0 + 12}{- 5 \cdot 0 + 15}$

Simplify $\frac{{(0)}^{2} - 7 \cdot 0 + 12}{- 5 \cdot 0 + 15}$ .

Simplify the numerator.

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

$y = \frac{0 - 7 \cdot 0 + 12}{- 5 \cdot 0 + 15}$

Multiply $- 7$ by $0$ .

$y = \frac{0 + 0 + 12}{- 5 \cdot 0 + 15}$

Add $0$ and $0$ .

$y = \frac{0 + 12}{- 5 \cdot 0 + 15}$

Add $0$ and $12$ .

$y = \frac{12}{- 5 \cdot 0 + 15}$

Simplify the denominator.

Multiply $- 5$ by $0$ .

$y = \frac{12}{0 + 15}$

Add $0$ and $15$ .

$y = \frac{12}{15}$

Cancel the common factor of $12$ and $15$ .

Factor $3$ out of $12$ .

$y = \frac{3 (4)}{15}$

Cancel the common factors.

Factor $3$ out of $15$ .

$y = \frac{3 \cdot 4}{3 \cdot 5}$

Cancel the common factor.

$y = \frac{3 \cdot 4}{3 \cdot 5}$

Rewrite the expression.

$y = \frac{4}{5}$

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{4}{5})$

List the intersections.

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

y-intercept(s): $(0, \frac{4}{5})$

Find the x and y Intercepts f(x)=(x^2-7x+12)/(-5x+15)

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