Find the X and Y Intercepts (2x^2+5x-12)/(3x^2+17x+20)

$\frac{(2 x^{2} + 5 x - 12)}{(3 x^{2} + 17 x + 20)}$

Write $\frac{(2 x^{2} + 5 x - 12)}{(3 x^{2} + 17 x + 20)}$ as an equation.

$y = \frac{(2 x^{2} + 5 x - 12)}{(3 x^{2} + 17 x + 20)}$

Find the x-intercepts.

Tap for more steps…

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

$0 = \frac{2 x^{2} + 5 x - 12}{3 x^{2} + 17 x + 20}$

Solve the equation.

Tap for more steps…

Rewrite the equation as $\frac{2 x^{2} + 5 x - 12}{3 x^{2} + 17 x + 20} = 0$ .

$\frac{2 x^{2} + 5 x - 12}{3 x^{2} + 17 x + 20} = 0$

Factor each term.

Tap for more steps…

Factor by grouping.

Tap for more steps…

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 $a \cdot c = 2 \cdot - 12 = - 24$ and whose sum is $b = 5$ .

Tap for more steps…

Factor $5$ out of $5 x$ .

$\frac{2 x^{2} + 5 (x) - 12}{3 x^{2} + 17 x + 20} = 0$

Rewrite $5$ as $- 3$ plus $8$

$\frac{2 x^{2} + (- 3 + 8) x - 12}{3 x^{2} + 17 x + 20} = 0$

Apply the distributive property.

$\frac{2 x^{2} - 3 x + 8 x - 12}{3 x^{2} + 17 x + 20} = 0$

Factor out the greatest common factor from each group.

Tap for more steps…

Group the first two terms and the last two terms.

$\frac{(2 x^{2} - 3 x) + 8 x - 12}{3 x^{2} + 17 x + 20} = 0$

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

$\frac{x (2 x - 3) + 4 (2 x - 3)}{3 x^{2} + 17 x + 20} = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x - 3$ .

$\frac{(2 x - 3) (x + 4)}{3 x^{2} + 17 x + 20} = 0$

Factor by grouping.

Tap for more steps…

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 $a \cdot c = 3 \cdot 20 = 60$ and whose sum is $b = 17$ .

Tap for more steps…

Factor $17$ out of $17 x$ .

$\frac{(2 x - 3) (x + 4)}{3 x^{2} + 17 (x) + 20} = 0$

Rewrite $17$ as $5$ plus $12$

$\frac{(2 x - 3) (x + 4)}{3 x^{2} + (5 + 12) x + 20} = 0$

Apply the distributive property.

$\frac{(2 x - 3) (x + 4)}{3 x^{2} + 5 x + 12 x + 20} = 0$

Factor out the greatest common factor from each group.

Tap for more steps…

Group the first two terms and the last two terms.

$\frac{(2 x - 3) (x + 4)}{(3 x^{2} + 5 x) + 12 x + 20} = 0$

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

$\frac{(2 x - 3) (x + 4)}{x (3 x + 5) + 4 (3 x + 5)} = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x + 5$ .

$\frac{(2 x - 3) (x + 4)}{(3 x + 5) (x + 4)} = 0$

Reduce the expression $\frac{(2 x - 3) (x + 4)}{(3 x + 5) (x + 4)}$ by cancelling the common factors.

Tap for more steps…

Cancel the common factor.

$\frac{(2 x - 3) (x + 4)}{(3 x + 5) (x + 4)} = 0$

Rewrite the expression.

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

Find the LCD of the terms in the equation.

Tap for more steps…

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3 x + 5, 1$

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

The factor for $3 x + 5$ is $3 x + 5$ itself.

$(3 x + 5) = 3 x + 5$

$(3 x + 5)$ occurs $1$ time.

The LCM of $3 x + 5$ is the result of multiplying all factors the greatest number of times they occur in either term.

$3 x + 5$

Multiply each term by $3 x + 5$ and simplify.

Tap for more steps…

Multiply each term in $\frac{2 x - 3}{3 x + 5} = 0$ by $3 x + 5$ in order to remove all the denominators from the equation.

$\frac{2 x - 3}{3 x + 5} \cdot (3 x + 5) = 0 \cdot (3 x + 5)$

Cancel the common factor of $3 x + 5$ .

Tap for more steps…

Cancel the common factor.

$\frac{2 x - 3}{3 x + 5} \cdot (3 x + 5) = 0 \cdot (3 x + 5)$

Rewrite the expression.

$2 x - 3 = 0 \cdot (3 x + 5)$

Multiply $0$ by $3 x + 5$ .

$2 x - 3 = 0$

Solve the equation.

Tap for more steps…

Add $3$ to both sides of the equation.

$2 x = 3$

Divide each term by $2$ and simplify.

Tap for more steps…

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Divide $x$ by $1$ .

$x = \frac{3}{2}$

x-intercept(s) in point form.

x-intercept(s): $(\frac{3}{2}, 0)$

Find the y-intercepts.

Tap for more steps…

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

$y = \frac{2 {(0)}^{2} + 5 (0) - 12}{3 {(0)}^{2} + 17 (0) + 20}$

Solve the equation.

Tap for more steps…

Remove parentheses.

$y = \frac{2 {(0)}^{2} + 5 (0) - 12}{3 {(0)}^{2} + 17 (0) + 20}$

Simplify $\frac{2 {(0)}^{2} + 5 (0) - 12}{3 {(0)}^{2} + 17 (0) + 20}$ .

Tap for more steps…

Simplify the numerator.

Tap for more steps…

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

$y = \frac{2 \cdot 0 + 5 (0) - 12}{3 {(0)}^{2} + 17 (0) + 20}$

Multiply $2$ by $0$ .

$y = \frac{0 + 5 (0) - 12}{3 {(0)}^{2} + 17 (0) + 20}$

Multiply $5$ by $0$ .

$y = \frac{0 + 0 - 12}{3 {(0)}^{2} + 17 (0) + 20}$

Add $0$ and $0$ .

$y = \frac{0 - 12}{3 {(0)}^{2} + 17 (0) + 20}$

Subtract $12$ from $0$ .

$y = \frac{- 12}{3 {(0)}^{2} + 17 (0) + 20}$

Simplify the denominator.

Tap for more steps…

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

$y = \frac{- 12}{3 \cdot 0 + 17 (0) + 20}$

Multiply $3$ by $0$ .

$y = \frac{- 12}{0 + 17 (0) + 20}$

Multiply $17$ by $0$ .

$y = \frac{- 12}{0 + 0 + 20}$

Add $0$ and $0$ .

$y = \frac{- 12}{0 + 20}$

Add $0$ and $20$ .

$y = \frac{- 12}{20}$

Reduce the expression by cancelling the common factors.

Tap for more steps…

Cancel the common factor of $- 12$ and $20$ .

Tap for more steps…

Factor $4$ out of $- 12$ .

$y = \frac{4 (- 3)}{20}$

Cancel the common factors.

Tap for more steps…

Factor $4$ out of $20$ .

$y = \frac{4 \cdot - 3}{4 \cdot 5}$

Cancel the common factor.

$y = \frac{4 \cdot - 3}{4 \cdot 5}$

Rewrite the expression.

$y = \frac{- 3}{5}$

Move the negative in front of the fraction.

$y = - \frac{3}{5}$

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{3}{5})$

List the intersections.

x-intercept(s): $(\frac{3}{2}, 0)$

y-intercept(s): $(0, - \frac{3}{5})$

Find the X and Y Intercepts (2x^2+5x-12)/(3x^2+17x+20)

Download our
App from the store

Create a High Performed UI/UX Design from a Silicon Valley.

Download our App from the store

Create a High Performed UI/UX Design from a Silicon Valley.

Download our
App from the store