Find the x and y Intercepts f(x)=(x^2+9x+8)/(5x^2+33x+18)

$f (x) = \frac{x^{2} + 9 x + 8}{5 x^{2} + 33 x + 18}$

Step 1

Find the x-intercepts.

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

$0 = \frac{x^{2} + 9 x + 8}{5 x^{2} + 33 x + 18}$

Solve the equation.

Set the numerator equal to zero.

$x^{2} + 9 x + 8 = 0$

Solve the equation for $x$ .

Factor $x^{2} + 9 x + 8$ 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 $8$ and whose sum is $9$ .

$1, 8$

Write the factored form using these integers.

$(x + 1) (x + 8) = 0$

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

$x + 1 = 0$

$x + 8 = 0$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set $x + 8$ equal to $0$ and solve for $x$ .

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

The final solution is all the values that make $(x + 1) (x + 8) = 0$ true.

$x = - 1, - 8$

x-intercept(s) in point form.

x-intercept(s): $(- 1, 0), (- 8, 0)$

Step 2

Find the y-intercepts.

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

$y = \frac{{(0)}^{2} + 9 (0) + 8}{5 {(0)}^{2} + 33 (0) + 18}$

Solve the equation.

Remove parentheses.

$y = \frac{0^{2} + 9 (0) + 8}{5 {(0)}^{2} + 33 (0) + 18}$

Remove parentheses.

$y = \frac{0^{2} + 9 (0) + 8}{5 \cdot 0^{2} + 33 (0) + 18}$

Remove parentheses.

$y = \frac{{(0)}^{2} + 9 (0) + 8}{5 {(0)}^{2} + 33 (0) + 18}$

Simplify $\frac{{(0)}^{2} + 9 (0) + 8}{5 {(0)}^{2} + 33 (0) + 18}$ .

Simplify the numerator.

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

$y = \frac{0 + 9 (0) + 8}{5 \cdot 0^{2} + 33 (0) + 18}$

Multiply $9$ by $0$ .

$y = \frac{0 + 0 + 8}{5 \cdot 0^{2} + 33 (0) + 18}$

Add $0$ and $0$ .

$y = \frac{0 + 8}{5 \cdot 0^{2} + 33 (0) + 18}$

Add $0$ and $8$ .

$y = \frac{8}{5 \cdot 0^{2} + 33 (0) + 18}$

Simplify the denominator.

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

$y = \frac{8}{5 \cdot 0 + 33 (0) + 18}$

Multiply $5$ by $0$ .

$y = \frac{8}{0 + 33 (0) + 18}$

Multiply $33$ by $0$ .

$y = \frac{8}{0 + 0 + 18}$

Add $0$ and $0$ .

$y = \frac{8}{0 + 18}$

Add $0$ and $18$ .

$y = \frac{8}{18}$

Cancel the common factor of $8$ and $18$ .

Factor $2$ out of $8$ .

$y = \frac{2 (4)}{18}$

Cancel the common factors.

Factor $2$ out of $18$ .

$y = \frac{2 \cdot 4}{2 \cdot 9}$

Cancel the common factor.

$y = \frac{2 \cdot 4}{2 \cdot 9}$

Rewrite the expression.

$y = \frac{4}{9}$

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{4}{9})$

Step 3

List the intersections.

x-intercept(s): $(- 1, 0), (- 8, 0)$

y-intercept(s): $(0, \frac{4}{9})$

Step 4

Find the x and y Intercepts f(x)=(x^2+9x+8)/(5x^2+33x+18)

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