Find the x and y Intercepts p(x)=2x^2+8x+2

$p (x) = 2 x^{2} + 8 x + 2$

Step 1

Find the x-intercepts.

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

$0 = 2 x^{2} + 8 x + 2$

Solve the equation.

Rewrite the equation as $2 x^{2} + 8 x + 2 = 0$ .

$2 x^{2} + 8 x + 2 = 0$

Factor $2$ out of $2 x^{2} + 8 x + 2$ .

Factor $2$ out of $2 x^{2}$ .

$2 (x^{2}) + 8 x + 2 = 0$

Factor $2$ out of $8 x$ .

$2 (x^{2}) + 2 (4 x) + 2 = 0$

Factor $2$ out of $2$ .

$2 (x^{2}) + 2 (4 x) + 2 (1) = 0$

Factor $2$ out of $2 (x^{2}) + 2 (4 x)$ .

$2 (x^{2} + 4 x) + 2 (1) = 0$

Factor $2$ out of $2 (x^{2} + 4 x) + 2 (1)$ .

$2 (x^{2} + 4 x + 1) = 0$

Divide each term in $2 (x^{2} + 4 x + 1) = 0$ by $2$ and simplify.

Divide each term in $2 (x^{2} + 4 x + 1) = 0$ by $2$ .

$\frac{2 (x^{2} + 4 x + 1)}{2} = \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 (x^{2} + 4 x + 1)}{2} = \frac{0}{2}$

Divide $x^{2} + 4 x + 1$ by $1$ .

$x^{2} + 4 x + 1 = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x^{2} + 4 x + 1 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = 4$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 1$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4}}{2 \cdot 1}$

Subtract $4$ from $16$ .

$x = \frac{- 4 \pm \sqrt{12}}{2 \cdot 1}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$x = \frac{- 4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

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

Simplify $\frac{- 4 \pm 2 \sqrt{3}}{2}$ .

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

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

Simplify the numerator.

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 1$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4}}{2 \cdot 1}$

Subtract $4$ from $16$ .

$x = \frac{- 4 \pm \sqrt{12}}{2 \cdot 1}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$x = \frac{- 4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

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

Simplify $\frac{- 4 \pm 2 \sqrt{3}}{2}$ .

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

Change the $\pm$ to $+$ .

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

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

Simplify the numerator.

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 1$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4}}{2 \cdot 1}$

Subtract $4$ from $16$ .

$x = \frac{- 4 \pm \sqrt{12}}{2 \cdot 1}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$x = \frac{- 4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

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

Simplify $\frac{- 4 \pm 2 \sqrt{3}}{2}$ .

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

Change the $\pm$ to $-$ .

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

The final answer is the combination of both solutions.

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

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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

$y = 2 {(0)}^{2} + 8 (0) + 2$

Solve the equation.

Remove parentheses.

$y = 2 \cdot 0^{2} + 8 (0) + 2$

Remove parentheses.

$y = 2 {(0)}^{2} + 8 (0) + 2$

Simplify $2 {(0)}^{2} + 8 (0) + 2$ .

Simplify each term.

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

$y = 2 \cdot 0 + 8 (0) + 2$

Multiply $2$ by $0$ .

$y = 0 + 8 (0) + 2$

Multiply $8$ 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): $(- 2 + \sqrt{3}, 0), (- 2 - \sqrt{3}, 0)$

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

Step 4

Find the x and y Intercepts p(x)=2x^2+8x+2

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