Find the x and y Intercepts f(x)=(x^2-7x-30)/(3x-21)

$f (x) = \frac{x^{2} - 7 x - 30}{3 x - 21}$

Step 1

Find the x-intercepts.

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

$0 = \frac{x^{2} - 7 x - 30}{3 x - 21}$

Solve the equation.

Set the numerator equal to zero.

$x^{2} - 7 x - 30 = 0$

Solve the equation for $x$ .

Factor $x^{2} - 7 x - 30$ 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 $- 30$ and whose sum is $- 7$ .

$- 10, 3$

Write the factored form using these integers.

$(x - 10) (x + 3) = 0$

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

$x - 10 = 0$

$x + 3 = 0$

Set $x - 10$ equal to $0$ and solve for $x$ .

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Add $10$ to both sides of the equation.

$x = 10$

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 $(x - 10) (x + 3) = 0$ true.

$x = 10, - 3$

x-intercept(s) in point form.

x-intercept(s): $(10, 0), (- 3, 0)$

Step 2

Find the y-intercepts.

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

$y = \frac{{(0)}^{2} - 7 \cdot 0 - 30}{3 (0) - 21}$

Solve the equation.

Remove parentheses.

$y = \frac{0^{2} - 7 \cdot 0 - 30}{3 (0) - 21}$

Remove parentheses.

$y = \frac{{(0)}^{2} - 7 \cdot 0 - 30}{3 (0) - 21}$

Simplify $\frac{{(0)}^{2} - 7 \cdot 0 - 30}{3 (0) - 21}$ .

Simplify the numerator.

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

$y = \frac{0 - 7 \cdot 0 - 30}{3 (0) - 21}$

Multiply $- 7$ by $0$ .

$y = \frac{0 + 0 - 30}{3 (0) - 21}$

Add $0$ and $0$ .

$y = \frac{0 - 30}{3 (0) - 21}$

Subtract $30$ from $0$ .

$y = \frac{- 30}{3 (0) - 21}$

Simplify the denominator.

Multiply $3$ by $0$ .

$y = \frac{- 30}{0 - 21}$

Subtract $21$ from $0$ .

$y = \frac{- 30}{- 21}$

Cancel the common factor of $- 30$ and $- 21$ .

Factor $- 3$ out of $- 30$ .

$y = \frac{- 3 (10)}{- 21}$

Cancel the common factors.

Factor $- 3$ out of $- 21$ .

$y = \frac{- 3 \cdot 10}{- 3 \cdot 7}$

Cancel the common factor.

$y = \frac{- 3 \cdot 10}{- 3 \cdot 7}$

Rewrite the expression.

$y = \frac{10}{7}$

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{10}{7})$

Step 3

List the intersections.

x-intercept(s): $(10, 0), (- 3, 0)$

y-intercept(s): $(0, \frac{10}{7})$

Step 4

Find the x and y Intercepts f(x)=(x^2-7x-30)/(3x-21)

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