Find the x and y Intercepts f(x)=-3x^2-4x+2

$f (x) = - 3 x^{2} - 4 x + 2$

Step 1

Find the x-intercepts.

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

$0 = - 3 x^{2} - 4 x + 2$

Solve the equation.

Rewrite the equation as $- 3 x^{2} - 4 x + 2 = 0$ .

$- 3 x^{2} - 4 x + 2 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 3$ , $b = - 4$ , and $c = 2$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 3 \cdot 2}}{2 \cdot - 3}$

Multiply $- 4 \cdot - 3 \cdot 2$ .

Multiply $- 4$ by $- 3$ .

$x = \frac{4 \pm \sqrt{16 + 12 \cdot 2}}{2 \cdot - 3}$

Multiply $12$ by $2$ .

$x = \frac{4 \pm \sqrt{16 + 24}}{2 \cdot - 3}$

Add $16$ and $24$ .

$x = \frac{4 \pm \sqrt{40}}{2 \cdot - 3}$

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

Factor $4$ out of $40$ .

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

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot - 3}$

Pull terms out from under the radical.

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

Multiply $2$ by $- 3$ .

$x = \frac{4 \pm 2 \sqrt{10}}{- 6}$

Simplify $\frac{4 \pm 2 \sqrt{10}}{- 6}$ .

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

Move the negative in front of the fraction.

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

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

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 3 \cdot 2}}{2 \cdot - 3}$

Multiply $- 4 \cdot - 3 \cdot 2$ .

Multiply $- 4$ by $- 3$ .

$x = \frac{4 \pm \sqrt{16 + 12 \cdot 2}}{2 \cdot - 3}$

Multiply $12$ by $2$ .

$x = \frac{4 \pm \sqrt{16 + 24}}{2 \cdot - 3}$

Add $16$ and $24$ .

$x = \frac{4 \pm \sqrt{40}}{2 \cdot - 3}$

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

Factor $4$ out of $40$ .

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

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot - 3}$

Pull terms out from under the radical.

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

Multiply $2$ by $- 3$ .

$x = \frac{4 \pm 2 \sqrt{10}}{- 6}$

Simplify $\frac{4 \pm 2 \sqrt{10}}{- 6}$ .

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

Move the negative in front of the fraction.

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

Change the $\pm$ to $+$ .

$x = - \frac{2 + \sqrt{10}}{3}$

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

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 3 \cdot 2}}{2 \cdot - 3}$

Multiply $- 4 \cdot - 3 \cdot 2$ .

Multiply $- 4$ by $- 3$ .

$x = \frac{4 \pm \sqrt{16 + 12 \cdot 2}}{2 \cdot - 3}$

Multiply $12$ by $2$ .

$x = \frac{4 \pm \sqrt{16 + 24}}{2 \cdot - 3}$

Add $16$ and $24$ .

$x = \frac{4 \pm \sqrt{40}}{2 \cdot - 3}$

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

Factor $4$ out of $40$ .

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

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot - 3}$

Pull terms out from under the radical.

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

Multiply $2$ by $- 3$ .

$x = \frac{4 \pm 2 \sqrt{10}}{- 6}$

Simplify $\frac{4 \pm 2 \sqrt{10}}{- 6}$ .

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

Move the negative in front of the fraction.

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

Change the $\pm$ to $-$ .

$x = - \frac{2 - \sqrt{10}}{3}$

The final answer is the combination of both solutions.

$x = - \frac{2 + \sqrt{10}}{3}, - \frac{2 - \sqrt{10}}{3}$

x-intercept(s) in point form.

x-intercept(s): $(- \frac{2 + \sqrt{10}}{3}, 0), (- \frac{2 - \sqrt{10}}{3}, 0)$

Step 2

Find the y-intercepts.

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

$y = - 3 {(0)}^{2} - 4 \cdot 0 + 2$

Solve the equation.

Remove parentheses.

$y = - 3 \cdot 0^{2} - 4 \cdot 0 + 2$

Remove parentheses.

$y = - 3 {(0)}^{2} - 4 \cdot 0 + 2$

Simplify $- 3 {(0)}^{2} - 4 \cdot 0 + 2$ .

Simplify each term.

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

$y = - 3 \cdot 0 - 4 \cdot 0 + 2$

Multiply $- 3$ by $0$ .

$y = 0 - 4 \cdot 0 + 2$

Multiply $- 4$ by $0$ .

$y = 0 + 0 + 2$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 2$

Add $0$ and $2$ .

$y = 2$

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(- \frac{2 + \sqrt{10}}{3}, 0), (- \frac{2 - \sqrt{10}}{3}, 0)$

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

Step 4

Find the x and y Intercepts f(x)=-3x^2-4x+2

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