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

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

Step 1

Find the x-intercepts.

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

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

Solve the equation.

Set the numerator equal to zero.

$x^{2} + 3 x = 0$

Solve the equation for $x$ .

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

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

$x \cdot x + 3 x = 0$

Factor $x$ out of $3 x$ .

$x \cdot x + x \cdot 3 = 0$

Factor $x$ out of $x \cdot x + x \cdot 3$ .

$x (x + 3) = 0$

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

$x = 0$

$x + 3 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x + 3$ equal to $0$ and solve for $x$ .

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

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

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

$x = 0, - 3$

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

$x = 0$

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

Remove parentheses.

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

Remove parentheses.

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

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

Simplify the numerator.

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

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

Multiply $3$ by $0$ .

$y = \frac{0 + 0}{0^{2} + 5 (0) + 6}$

Add $0$ and $0$ .

$y = \frac{0}{0^{2} + 5 (0) + 6}$

Simplify the denominator.

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

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

Multiply $5$ by $0$ .

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

Add $0$ and $0$ .

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

Add $0$ and $6$ .

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

Divide $0$ by $6$ .

$y = 0$

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4

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

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