Find the x and y Intercepts 2x^2+x-3

$2 x^{2} + x - 3$

Step 1

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

$y = 2 x^{2} + x - 3$

Step 2

Find the x-intercepts.

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

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

Solve the equation.

Rewrite the equation as $2 x^{2} + x - 3 = 0$ .

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

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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = 1$ .

Multiply by $1$ .

$2 x^{2} + 1 x - 3 = 0$

Rewrite $1$ as $- 2$ plus $3$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$2 x (x - 1) + 3 (x - 1) = 0$

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

$(x - 1) (2 x + 3) = 0$

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

$x - 1 = 0$

$2 x + 3 = 0$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $2 x + 3$ equal to $0$ and solve for $x$ .

Set $2 x + 3$ equal to $0$ .

$2 x + 3 = 0$

Solve $2 x + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Divide each term in $2 x = - 3$ by $2$ and simplify.

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

The final solution is all the values that make $(x - 1) (2 x + 3) = 0$ true.

$x = 1, - \frac{3}{2}$

x-intercept(s) in point form.

x-intercept(s): $(1, 0), (- \frac{3}{2}, 0)$

Step 3

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

Remove parentheses.

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

Remove parentheses.

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

Simplify $2 {(0)}^{2} + 0 - 3$ .

Simplify each term.

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

$y = 2 \cdot 0 + 0 - 3$

Multiply $2$ by $0$ .

$y = 0 + 0 - 3$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$y = 0 - 3$

Subtract $3$ from $0$ .

$y = - 3$

y-intercept(s) in point form.

y-intercept(s): $(0, - 3)$

Step 4

List the intersections.

x-intercept(s): $(1, 0), (- \frac{3}{2}, 0)$

y-intercept(s): $(0, - 3)$

Step 5

Find the x and y Intercepts 2x^2+x-3

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