Find the x and y Intercepts (2x^2-3x-2)/(x-2)

$\frac{2 x^{2} - 3 x - 2}{x - 2}$

Step 1

Write $\frac{2 x^{2} - 3 x - 2}{x - 2}$ as an equation.

$y = \frac{2 x^{2} - 3 x - 2}{x - 2}$

Step 2

Find the x-intercepts.

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

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

Solve the equation.

Set the numerator equal to zero.

$2 x^{2} - 3 x - 2 = 0$

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 $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = - 3$ .

Factor $- 3$ out of $- 3 x$ .

$2 x^{2} - 3 x - 2 = 0$

Rewrite $- 3$ as $1$ plus $- 4$

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

Apply the distributive property.

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

Multiply $x$ by $1$ .

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

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

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

$2 x + 1 = 0$

$x - 2 = 0$

Set $2 x + 1$ equal to $0$ and solve for $x$ .

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Solve $2 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Divide each term in $2 x = - 1$ by $2$ and simplify.

Divide each term in $2 x = - 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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 $(2 x + 1) (x - 2) = 0$ true.

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

Exclude the solutions that do not make $0 = \frac{2 x^{2} - 3 x - 2}{x - 2}$ true.

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

x-intercept(s) in point form.

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

Step 3

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

Remove parentheses.

$y = \frac{2 \cdot 0^{2} - 3 \cdot 0 - 2}{0 - 2}$

Remove parentheses.

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

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

Cancel the common factor of $2 {(0)}^{2} - 3 \cdot 0 - 2$ and $(0) - 2$ .

Factor $2$ out of $2 {(0)}^{2}$ .

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

Factor $2$ out of $- 3 \cdot 0$ .

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

Factor $2$ out of $2 ({(0)}^{2}) + 2 (- 3 \cdot 0)$ .

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

Factor $2$ out of $- 2$ .

$y = \frac{2 ({(0)}^{2} - 3 \cdot 0) + 2 (- 1)}{(0) - 2}$

Factor $2$ out of $2 ({(0)}^{2} - 3 \cdot 0) + 2 (- 1)$ .

$y = \frac{2 ({(0)}^{2} - 3 \cdot 0 - 1)}{(0) - 2}$

Cancel the common factors.

Factor $2$ out of $0$ .

$y = \frac{2 ({(0)}^{2} - 3 \cdot 0 - 1)}{2 (0) - 2}$

Factor $2$ out of $- 2$ .

$y = \frac{2 ({(0)}^{2} - 3 \cdot 0 - 1)}{2 \cdot 0 + 2 \cdot - 1}$

Factor $2$ out of $2 \cdot 0 + 2 \cdot - 1$ .

$y = \frac{2 ({(0)}^{2} - 3 \cdot 0 - 1)}{2 \cdot (0 - 1)}$

Cancel the common factor.

$y = \frac{2 ({(0)}^{2} - 3 \cdot 0 - 1)}{2 \cdot (0 - 1)}$

Rewrite the expression.

$y = \frac{{(0)}^{2} - 3 \cdot 0 - 1}{0 - 1}$

Simplify the numerator.

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

$y = \frac{0 - 3 \cdot 0 - 1}{0 - 1}$

Multiply $- 3$ by $0$ .

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

Add $0$ and $0$ .

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

Subtract $1$ from $0$ .

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

Simplify the expression.

Subtract $1$ from $0$ .

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

Divide $- 1$ by $- 1$ .

$y = 1$

y-intercept(s) in point form.

y-intercept(s): $(0, 1)$

Step 4

List the intersections.

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

y-intercept(s): $(0, 1)$

Step 5

Find the x and y Intercepts (2x^2-3x-2)/(x-2)

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