Find the x and y Intercepts f(x)=-2x-2+9x^2

$f (x) = - 2 x - 2 + 9 x^{2}$

Step 1

Find the x-intercepts.

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

$0 = - 2 x - 2 + 9 x^{2}$

Solve the equation.

Rewrite the equation as $- 2 x - 2 + 9 x^{2} = 0$ .

$- 2 x - 2 + 9 x^{2} = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 9$ , $b = - 2$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (9 \cdot - 2)}}{2 \cdot 9}$

Simplify.

Simplify the numerator.

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 9 \cdot - 2}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot - 2$ .

Multiply $- 4$ by $9$ .

$x = \frac{2 \pm \sqrt{4 - 36 \cdot - 2}}{2 \cdot 9}$

Multiply $- 36$ by $- 2$ .

$x = \frac{2 \pm \sqrt{4 + 72}}{2 \cdot 9}$

Add $4$ and $72$ .

$x = \frac{2 \pm \sqrt{76}}{2 \cdot 9}$

Rewrite $76$ as $2^{2} \cdot 19$ .

Factor $4$ out of $76$ .

$x = \frac{2 \pm \sqrt{4 (19)}}{2 \cdot 9}$

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 9}$

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{19}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{2 \pm 2 \sqrt{19}}{18}$

Simplify $\frac{2 \pm 2 \sqrt{19}}{18}$ .

$x = \frac{1 \pm \sqrt{19}}{9}$

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

Simplify the numerator.

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 9 \cdot - 2}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot - 2$ .

Multiply $- 4$ by $9$ .

$x = \frac{2 \pm \sqrt{4 - 36 \cdot - 2}}{2 \cdot 9}$

Multiply $- 36$ by $- 2$ .

$x = \frac{2 \pm \sqrt{4 + 72}}{2 \cdot 9}$

Add $4$ and $72$ .

$x = \frac{2 \pm \sqrt{76}}{2 \cdot 9}$

Rewrite $76$ as $2^{2} \cdot 19$ .

Factor $4$ out of $76$ .

$x = \frac{2 \pm \sqrt{4 (19)}}{2 \cdot 9}$

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 9}$

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{19}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{2 \pm 2 \sqrt{19}}{18}$

Simplify $\frac{2 \pm 2 \sqrt{19}}{18}$ .

$x = \frac{1 \pm \sqrt{19}}{9}$

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{19}}{9}$

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

Simplify the numerator.

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 9 \cdot - 2}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot - 2$ .

Multiply $- 4$ by $9$ .

$x = \frac{2 \pm \sqrt{4 - 36 \cdot - 2}}{2 \cdot 9}$

Multiply $- 36$ by $- 2$ .

$x = \frac{2 \pm \sqrt{4 + 72}}{2 \cdot 9}$

Add $4$ and $72$ .

$x = \frac{2 \pm \sqrt{76}}{2 \cdot 9}$

Rewrite $76$ as $2^{2} \cdot 19$ .

Factor $4$ out of $76$ .

$x = \frac{2 \pm \sqrt{4 (19)}}{2 \cdot 9}$

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 9}$

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{19}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{2 \pm 2 \sqrt{19}}{18}$

Simplify $\frac{2 \pm 2 \sqrt{19}}{18}$ .

$x = \frac{1 \pm \sqrt{19}}{9}$

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{19}}{9}$

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{19}}{9}, \frac{1 - \sqrt{19}}{9}$

x-intercept(s) in point form.

x-intercept(s): $(\frac{1 + \sqrt{19}}{9}, 0), (\frac{1 - \sqrt{19}}{9}, 0)$

Step 2

Find the y-intercepts.

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

$y = - 2 \cdot 0 - 2 + 9 {(0)}^{2}$

Solve the equation.

Remove parentheses.

$y = - 2 \cdot 0 - 2 + 9 \cdot 0^{2}$

Remove parentheses.

$y = - 2 \cdot 0 - 2 + 9 {(0)}^{2}$

Simplify $- 2 \cdot 0 - 2 + 9 {(0)}^{2}$ .

Simplify each term.

Multiply $- 2$ by $0$ .

$y = 0 - 2 + 9 {(0)}^{2}$

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

$y = 0 - 2 + 9 \cdot 0$

Multiply $9$ by $0$ .

$y = 0 - 2 + 0$

Simplify by adding and subtracting.

Subtract $2$ from $0$ .

$y = - 2 + 0$

Add $- 2$ and $0$ .

$y = - 2$

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(\frac{1 + \sqrt{19}}{9}, 0), (\frac{1 - \sqrt{19}}{9}, 0)$

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

Step 4

Find the x and y Intercepts f(x)=-2x-2+9x^2

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