Find the x and y Intercepts f(x)=(3x^2-19x+28)/(x^2-36)

$f (x) = \frac{3 x^{2} - 19 x + 28}{x^{2} - 36}$

Find the x-intercepts.

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

$0 = \frac{3 x^{2} - 19 x + 28}{x^{2} - 36}$

Solve the equation.

Set the numerator equal to zero.

$3 x^{2} - 19 x + 28 = 0$

Solve the equation for $x$ .

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 28 = 84$ and whose sum is $b = - 19$ .

Factor $- 19$ out of $- 19 x$ .

$3 x^{2} - 19 x + 28 = 0$

Rewrite $- 19$ as $- 7$ plus $- 12$

$3 x^{2} + (- 7 - 12) x + 28 = 0$

Apply the distributive property.

$3 x^{2} - 7 x - 12 x + 28 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x^{2} - 7 x) - 12 x + 28 = 0$

Factor out the greatest common factor (GCF) from each group.

$x (3 x - 7) - 4 (3 x - 7) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x - 7$ .

$(3 x - 7) (x - 4) = 0$

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

$3 x - 7 = 0$

$x - 4 = 0$

Set $3 x - 7$ equal to $0$ and solve for $x$ .

Set $3 x - 7$ equal to $0$ .

$3 x - 7 = 0$

Solve $3 x - 7 = 0$ for $x$ .

Add $7$ to both sides of the equation.

$3 x = 7$

Divide each term in $3 x = 7$ by $3$ and simplify.

Divide each term in $3 x = 7$ by $3$ .

$\frac{3 x}{3} = \frac{7}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{7}{3}$

Divide $x$ by $1$ .

$x = \frac{7}{3}$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make $(3 x - 7) (x - 4) = 0$ true.

$x = \frac{7}{3}, 4$

x-intercept(s) in point form.

x-intercept(s): $(\frac{7}{3}, 0), (4, 0)$

Find the y-intercepts.

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

$y = \frac{3 {(0)}^{2} - 19 \cdot 0 + 28}{{(0)}^{2} - 36}$

Solve the equation.

Remove parentheses.

$y = \frac{3 \cdot 0^{2} - 19 \cdot 0 + 28}{{(0)}^{2} - 36}$

Remove parentheses.

$y = \frac{3 \cdot 0^{2} - 19 \cdot 0 + 28}{0^{2} - 36}$

Remove parentheses.

$y = \frac{3 {(0)}^{2} - 19 \cdot 0 + 28}{{(0)}^{2} - 36}$

Simplify $\frac{3 {(0)}^{2} - 19 \cdot 0 + 28}{{(0)}^{2} - 36}$ .

Simplify the numerator.

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

$y = \frac{3 \cdot 0 - 19 \cdot 0 + 28}{0^{2} - 36}$

Multiply $3$ by $0$ .

$y = \frac{0 - 19 \cdot 0 + 28}{0^{2} - 36}$

Multiply $- 19$ by $0$ .

$y = \frac{0 + 0 + 28}{0^{2} - 36}$

Add $0$ and $0$ .

$y = \frac{0 + 28}{0^{2} - 36}$

Add $0$ and $28$ .

$y = \frac{28}{0^{2} - 36}$

Simplify the denominator.

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

$y = \frac{28}{0 - 36}$

Subtract $36$ from $0$ .

$y = \frac{28}{- 36}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $28$ and $- 36$ .

Factor $4$ out of $28$ .

$y = \frac{4 (7)}{- 36}$

Cancel the common factors.

Factor $4$ out of $- 36$ .

$y = \frac{4 \cdot 7}{4 \cdot - 9}$

Cancel the common factor.

$y = \frac{4 \cdot 7}{4 \cdot - 9}$

Rewrite the expression.

$y = \frac{7}{- 9}$

Move the negative in front of the fraction.

$y = - \frac{7}{9}$

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{7}{9})$

List the intersections.

x-intercept(s): $(\frac{7}{3}, 0), (4, 0)$

y-intercept(s): $(0, - \frac{7}{9})$

Find the x and y Intercepts f(x)=(3x^2-19x+28)/(x^2-36)

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