Find the x and y Intercepts f(x)=(x^2+x-6)/(x^2+x-20)

$f (x) = \frac{x^{2} + x - 6}{x^{2} + x - 20}$

Step 1

Find the x-intercepts.

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

$0 = \frac{x^{2} + x - 6}{x^{2} + x - 20}$

Solve the equation.

Set the numerator equal to zero.

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

Solve the equation for $x$ .

Factor $x^{2} + x - 6$ 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 $1$ .

$- 2, 3$

Write the factored form using these integers.

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

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

$x - 2 = 0$

$x + 3 = 0$

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 $x + 3$ 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 $(x - 2) (x + 3) = 0$ true.

$x = 2, - 3$

x-intercept(s) in point form.

x-intercept(s): $(2, 0), (- 3, 0)$

Step 2

Find the y-intercepts.

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

$y = \frac{{(0)}^{2} + 0 - 6}{{(0)}^{2} + 0 - 20}$

Solve the equation.

Remove parentheses.

$y = \frac{{(0)}^{2} + 0 - 6}{{(0)}^{2} + 0 - 20}$

Remove parentheses.

$y = \frac{0^{2} + 0 - 6}{{(0)}^{2} + 0 - 20}$

Remove parentheses.

$y = \frac{0^{2} + 0 - 6}{0^{2} + 0 - 20}$

Remove parentheses.

$y = \frac{{(0)}^{2} + 0 - 6}{{(0)}^{2} + 0 - 20}$

Simplify $\frac{{(0)}^{2} + 0 - 6}{{(0)}^{2} + 0 - 20}$ .

Simplify the numerator.

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

$y = \frac{0 + 0 - 6}{0^{2} + 0 - 20}$

Add $0$ and $0$ .

$y = \frac{0 - 6}{0^{2} + 0 - 20}$

Subtract $6$ from $0$ .

$y = \frac{- 6}{0^{2} + 0 - 20}$

Simplify the denominator.

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

$y = \frac{- 6}{0 + 0 - 20}$

Add $0$ and $0$ .

$y = \frac{- 6}{0 - 20}$

Subtract $20$ from $0$ .

$y = \frac{- 6}{- 20}$

Cancel the common factor of $- 6$ and $- 20$ .

Factor $- 2$ out of $- 6$ .

$y = \frac{- 2 (3)}{- 20}$

Cancel the common factors.

Factor $- 2$ out of $- 20$ .

$y = \frac{- 2 \cdot 3}{- 2 \cdot 10}$

Cancel the common factor.

$y = \frac{- 2 \cdot 3}{- 2 \cdot 10}$

Rewrite the expression.

$y = \frac{3}{10}$

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{3}{10})$

Step 3

List the intersections.

x-intercept(s): $(2, 0), (- 3, 0)$

y-intercept(s): $(0, \frac{3}{10})$

Step 4

Find the x and y Intercepts f(x)=(x^2+x-6)/(x^2+x-20)

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