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

$f (x) = 3 x^{2} - 2 x - 3$

Step 1

Find the x-intercepts.

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

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

Solve the equation.

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

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

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 3$ , $b = - 2$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (3 \cdot - 3)}}{2 \cdot 3}$

Simplify.

Simplify the numerator.

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 3 \cdot - 3}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 3$ .

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot - 3}}{2 \cdot 3}$

Multiply $- 12$ by $- 3$ .

$x = \frac{2 \pm \sqrt{4 + 36}}{2 \cdot 3}$

Add $4$ and $36$ .

$x = \frac{2 \pm \sqrt{40}}{2 \cdot 3}$

Rewrite $40$ as $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

$x = \frac{2 \pm \sqrt{4 (10)}}{2 \cdot 3}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{10}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{2 \pm 2 \sqrt{10}}{6}$

Simplify $\frac{2 \pm 2 \sqrt{10}}{6}$ .

$x = \frac{1 \pm \sqrt{10}}{3}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 3 \cdot - 3}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 3$ .

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot - 3}}{2 \cdot 3}$

Multiply $- 12$ by $- 3$ .

$x = \frac{2 \pm \sqrt{4 + 36}}{2 \cdot 3}$

Add $4$ and $36$ .

$x = \frac{2 \pm \sqrt{40}}{2 \cdot 3}$

Rewrite $40$ as $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

$x = \frac{2 \pm \sqrt{4 (10)}}{2 \cdot 3}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{10}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{2 \pm 2 \sqrt{10}}{6}$

Simplify $\frac{2 \pm 2 \sqrt{10}}{6}$ .

$x = \frac{1 \pm \sqrt{10}}{3}$

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{10}}{3}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 3 \cdot - 3}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 3$ .

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot - 3}}{2 \cdot 3}$

Multiply $- 12$ by $- 3$ .

$x = \frac{2 \pm \sqrt{4 + 36}}{2 \cdot 3}$

Add $4$ and $36$ .

$x = \frac{2 \pm \sqrt{40}}{2 \cdot 3}$

Rewrite $40$ as $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

$x = \frac{2 \pm \sqrt{4 (10)}}{2 \cdot 3}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{10}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{2 \pm 2 \sqrt{10}}{6}$

Simplify $\frac{2 \pm 2 \sqrt{10}}{6}$ .

$x = \frac{1 \pm \sqrt{10}}{3}$

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{10}}{3}$

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{10}}{3}, \frac{1 - \sqrt{10}}{3}$

x-intercept(s) in point form.

x-intercept(s): $(\frac{1 + \sqrt{10}}{3}, 0), (\frac{1 - \sqrt{10}}{3}, 0)$

Step 2

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

Remove parentheses.

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

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

Simplify each term.

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

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

Multiply $3$ by $0$ .

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

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 3

List the intersections.

x-intercept(s): $(\frac{1 + \sqrt{10}}{3}, 0), (\frac{1 - \sqrt{10}}{3}, 0)$

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

Step 4

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

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