Find the x and y Intercepts (x^2-5x+6)/(x^2-4x+3)

$\frac{x^{2} - 5 x + 6}{x^{2} - 4 x + 3}$

Step 1

Write $\frac{x^{2} - 5 x + 6}{x^{2} - 4 x + 3}$ as an equation.

$y = \frac{x^{2} - 5 x + 6}{x^{2} - 4 x + 3}$

Step 2

Find the x-intercepts.

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

$0 = \frac{x^{2} - 5 x + 6}{x^{2} - 4 x + 3}$

Solve the equation.

Set the numerator equal to zero.

$x^{2} - 5 x + 6 = 0$

Solve the equation for $x$ .

Factor $x^{2} - 5 x + 6$ 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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Write the factored form using these integers.

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

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

$x - 3 = 0$

$x - 2 = 0$

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.

$x = 3$

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.

$x = 2$

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

$x = 3, 2$

Exclude the solutions that do not make $0 = \frac{x^{2} - 5 x + 6}{x^{2} - 4 x + 3}$ true.

$x = 2$

x-intercept(s) in point form.

x-intercept(s): $(2, 0)$

Step 3

Find the y-intercepts.

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

$y = \frac{{(0)}^{2} - 5 \cdot 0 + 6}{{(0)}^{2} - 4 \cdot 0 + 3}$

Solve the equation.

Remove parentheses.

$y = \frac{0^{2} - 5 \cdot 0 + 6}{{(0)}^{2} - 4 \cdot 0 + 3}$

Remove parentheses.

$y = \frac{0^{2} - 5 \cdot 0 + 6}{0^{2} - 4 \cdot 0 + 3}$

Remove parentheses.

$y = \frac{{(0)}^{2} - 5 \cdot 0 + 6}{{(0)}^{2} - 4 \cdot 0 + 3}$

Simplify $\frac{{(0)}^{2} - 5 \cdot 0 + 6}{{(0)}^{2} - 4 \cdot 0 + 3}$ .

Simplify the numerator.

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

$y = \frac{0 - 5 \cdot 0 + 6}{0^{2} - 4 \cdot 0 + 3}$

Multiply $- 5$ by $0$ .

$y = \frac{0 + 0 + 6}{0^{2} - 4 \cdot 0 + 3}$

Add $0$ and $0$ .

$y = \frac{0 + 6}{0^{2} - 4 \cdot 0 + 3}$

Add $0$ and $6$ .

$y = \frac{6}{0^{2} - 4 \cdot 0 + 3}$

Simplify the denominator.

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

$y = \frac{6}{0 - 4 \cdot 0 + 3}$

Multiply $- 4$ by $0$ .

$y = \frac{6}{0 + 0 + 3}$

Add $0$ and $0$ .

$y = \frac{6}{0 + 3}$

Add $0$ and $3$ .

$y = \frac{6}{3}$

Divide $6$ by $3$ .

$y = 2$

y-intercept(s) in point form.

y-intercept(s): $(0, 2)$

Step 4

List the intersections.

x-intercept(s): $(2, 0)$

y-intercept(s): $(0, 2)$

Step 5

Find the x and y Intercepts (x^2-5x+6)/(x^2-4x+3)

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