Find the x and y Intercepts (4x^2-7x-2)/(x^2+1)

$\frac{4 x^{2} - 7 x - 2}{x^{2} + 1}$

Step 1

Write $\frac{4 x^{2} - 7 x - 2}{x^{2} + 1}$ as an equation.

$y = \frac{4 x^{2} - 7 x - 2}{x^{2} + 1}$

Step 2

Find the x-intercepts.

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

$0 = \frac{4 x^{2} - 7 x - 2}{x^{2} + 1}$

Solve the equation.

Set the numerator equal to zero.

$4 x^{2} - 7 x - 2 = 0$

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 $a \cdot c = 4 \cdot - 2 = - 8$ and whose sum is $b = - 7$ .

Factor $- 7$ out of $- 7 x$ .

$4 x^{2} - 7 x - 2 = 0$

Rewrite $- 7$ as $1$ plus $- 8$

$4 x^{2} + (1 - 8) x - 2 = 0$

Apply the distributive property.

$4 x^{2} + 1 x - 8 x - 2 = 0$

Multiply $x$ by $1$ .

$4 x^{2} + x - 8 x - 2 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 x^{2} + x) - 8 x - 2 = 0$

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

$x (4 x + 1) - 2 (4 x + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

$(4 x + 1) (x - 2) = 0$

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

$4 x + 1 = 0$

$x - 2 = 0$

Set $4 x + 1$ equal to $0$ and solve for $x$ .

Set $4 x + 1$ equal to $0$ .

$4 x + 1 = 0$

Solve $4 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Divide each term in $4 x = - 1$ by $4$ and simplify.

Divide each term in $4 x = - 1$ by $4$ .

$\frac{4 x}{4} = \frac{- 1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 1}{4}$

Divide $x$ by $1$ .

$x = \frac{- 1}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

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.

$x = 2$

The final solution is all the values that make $(4 x + 1) (x - 2) = 0$ true.

$x = - \frac{1}{4}, 2$

x-intercept(s) in point form.

x-intercept(s): $(- \frac{1}{4}, 0), (2, 0)$

Step 3

Find the y-intercepts.

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

$y = \frac{4 {(0)}^{2} - 7 \cdot 0 - 2}{{(0)}^{2} + 1}$

Solve the equation.

Remove parentheses.

$y = \frac{4 \cdot 0^{2} - 7 \cdot 0 - 2}{{(0)}^{2} + 1}$

Remove parentheses.

$y = \frac{4 \cdot 0^{2} - 7 \cdot 0 - 2}{0^{2} + 1}$

Remove parentheses.

$y = \frac{4 {(0)}^{2} - 7 \cdot 0 - 2}{{(0)}^{2} + 1}$

Simplify $\frac{4 {(0)}^{2} - 7 \cdot 0 - 2}{{(0)}^{2} + 1}$ .

Simplify the numerator.

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

$y = \frac{4 \cdot 0 - 7 \cdot 0 - 2}{0^{2} + 1}$

Multiply $4$ by $0$ .

$y = \frac{0 - 7 \cdot 0 - 2}{0^{2} + 1}$

Multiply $- 7$ by $0$ .

$y = \frac{0 + 0 - 2}{0^{2} + 1}$

Add $0$ and $0$ .

$y = \frac{0 - 2}{0^{2} + 1}$

Subtract $2$ from $0$ .

$y = \frac{- 2}{0^{2} + 1}$

Simplify the denominator.

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

$y = \frac{- 2}{0 + 1}$

Add $0$ and $1$ .

$y = \frac{- 2}{1}$

Divide $- 2$ by $1$ .

$y = - 2$

y-intercept(s) in point form.

y-intercept(s): $(0, - 2)$

Step 4

List the intersections.

x-intercept(s): $(- \frac{1}{4}, 0), (2, 0)$

y-intercept(s): $(0, - 2)$

Step 5

Find the x and y Intercepts (4x^2-7x-2)/(x^2+1)

Related Questions

Support LCMGCF.com by disabling your adblocker.