Find the x and y Intercepts y=(2x^2+5x-3)/(x+3)

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

Step 1

Find the x-intercepts.

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

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

Solve the equation.

Set the numerator equal to zero.

$2 x^{2} + 5 x - 3 = 0$

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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = 5$ .

Factor $5$ out of $5 x$ .

$2 x^{2} + 5 (x) - 3 = 0$

Rewrite $5$ as $- 1$ plus $6$

$2 x^{2} + (- 1 + 6) x - 3 = 0$

Apply the distributive property.

$2 x^{2} - 1 x + 6 x - 3 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 x^{2} - 1 x) + 6 x - 3 = 0$

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

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

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

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

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

$2 x - 1 = 0$

$x + 3 = 0$

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 $x$ .

Add $1$ to 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 $1$ .

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

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

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

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

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

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

Exclude the solutions that do not make $0 = \frac{2 x^{2} + 5 x - 3}{x + 3}$ true.

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

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

Remove parentheses.

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

Remove parentheses.

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

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

Simplify the numerator.

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

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

Multiply $2$ by $0$ .

$y = \frac{0 + 5 (0) - 3}{0 + 3}$

Multiply $5$ by $0$ .

$y = \frac{0 + 0 - 3}{0 + 3}$

Add $0$ and $0$ .

$y = \frac{0 - 3}{0 + 3}$

Subtract $3$ from $0$ .

$y = \frac{- 3}{0 + 3}$

Simplify the expression.

Add $0$ and $3$ .

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

Divide $- 3$ by $3$ .

$y = - 1$

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4

Find the x and y Intercepts y=(2x^2+5x-3)/(x+3)

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