Find the x and y Intercepts 4x^2+8x+9

$4 x^{2} + 8 x + 9$

Step 1

Write $4 x^{2} + 8 x + 9$ as an equation.

$y = 4 x^{2} + 8 x + 9$

Step 2

Find the x-intercepts.

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

$0 = 4 x^{2} + 8 x + 9$

Solve the equation.

Rewrite the equation as $4 x^{2} + 8 x + 9 = 0$ .

$4 x^{2} + 8 x + 9 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 4$ , $b = 8$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (4 \cdot 9)}}{2 \cdot 4}$

Simplify.

Simplify the numerator.

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 4 \cdot 9}}{2 \cdot 4}$

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

Multiply $- 4$ by $4$ .

$x = \frac{- 8 \pm \sqrt{64 - 16 \cdot 9}}{2 \cdot 4}$

Multiply $- 16$ by $9$ .

$x = \frac{- 8 \pm \sqrt{64 - 144}}{2 \cdot 4}$

Subtract $144$ from $64$ .

$x = \frac{- 8 \pm \sqrt{- 80}}{2 \cdot 4}$

Rewrite $- 80$ as $- 1 (80)$ .

$x = \frac{- 8 \pm \sqrt{- 1 \cdot 80}}{2 \cdot 4}$

Rewrite $\sqrt{- 1 (80)}$ as $\sqrt{- 1} \cdot \sqrt{80}$ .

$x = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{80}}{2 \cdot 4}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 8 \pm i \cdot \sqrt{80}}{2 \cdot 4}$

Rewrite $80$ as $4^{2} \cdot 5$ .

Factor $16$ out of $80$ .

$x = \frac{- 8 \pm i \cdot \sqrt{16 (5)}}{2 \cdot 4}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 8 \pm i \cdot \sqrt{4^{2} \cdot 5}}{2 \cdot 4}$

Pull terms out from under the radical.

$x = \frac{- 8 \pm i \cdot (4 \sqrt{5})}{2 \cdot 4}$

Move $4$ to the left of $i$ .

$x = \frac{- 8 \pm 4 i \sqrt{5}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{- 8 \pm 4 i \sqrt{5}}{8}$

Simplify $\frac{- 8 \pm 4 i \sqrt{5}}{8}$ .

$x = \frac{- 2 \pm i \sqrt{5}}{2}$

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

Simplify the numerator.

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 4 \cdot 9}}{2 \cdot 4}$

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

Multiply $- 4$ by $4$ .

$x = \frac{- 8 \pm \sqrt{64 - 16 \cdot 9}}{2 \cdot 4}$

Multiply $- 16$ by $9$ .

$x = \frac{- 8 \pm \sqrt{64 - 144}}{2 \cdot 4}$

Subtract $144$ from $64$ .

$x = \frac{- 8 \pm \sqrt{- 80}}{2 \cdot 4}$

Rewrite $- 80$ as $- 1 (80)$ .

$x = \frac{- 8 \pm \sqrt{- 1 \cdot 80}}{2 \cdot 4}$

Rewrite $\sqrt{- 1 (80)}$ as $\sqrt{- 1} \cdot \sqrt{80}$ .

$x = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{80}}{2 \cdot 4}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 8 \pm i \cdot \sqrt{80}}{2 \cdot 4}$

Rewrite $80$ as $4^{2} \cdot 5$ .

Factor $16$ out of $80$ .

$x = \frac{- 8 \pm i \cdot \sqrt{16 (5)}}{2 \cdot 4}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 8 \pm i \cdot \sqrt{4^{2} \cdot 5}}{2 \cdot 4}$

Pull terms out from under the radical.

$x = \frac{- 8 \pm i \cdot (4 \sqrt{5})}{2 \cdot 4}$

Move $4$ to the left of $i$ .

$x = \frac{- 8 \pm 4 i \sqrt{5}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{- 8 \pm 4 i \sqrt{5}}{8}$

Simplify $\frac{- 8 \pm 4 i \sqrt{5}}{8}$ .

$x = \frac{- 2 \pm i \sqrt{5}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{- 2 + i \sqrt{5}}{2}$

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- 1 \cdot 2 + i \sqrt{5}}{2}$

Factor $- 1$ out of $i \sqrt{5}$ .

$x = \frac{- 1 \cdot 2 - (- i \sqrt{5})}{2}$

Factor $- 1$ out of $- 1 (2) - (- i \sqrt{5})$ .

$x = \frac{- 1 (2 - i \sqrt{5})}{2}$

Move the negative in front of the fraction.

$x = - \frac{2 - i \sqrt{5}}{2}$

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

Simplify the numerator.

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 4 \cdot 9}}{2 \cdot 4}$

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

Multiply $- 4$ by $4$ .

$x = \frac{- 8 \pm \sqrt{64 - 16 \cdot 9}}{2 \cdot 4}$

Multiply $- 16$ by $9$ .

$x = \frac{- 8 \pm \sqrt{64 - 144}}{2 \cdot 4}$

Subtract $144$ from $64$ .

$x = \frac{- 8 \pm \sqrt{- 80}}{2 \cdot 4}$

Rewrite $- 80$ as $- 1 (80)$ .

$x = \frac{- 8 \pm \sqrt{- 1 \cdot 80}}{2 \cdot 4}$

Rewrite $\sqrt{- 1 (80)}$ as $\sqrt{- 1} \cdot \sqrt{80}$ .

$x = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{80}}{2 \cdot 4}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 8 \pm i \cdot \sqrt{80}}{2 \cdot 4}$

Rewrite $80$ as $4^{2} \cdot 5$ .

Factor $16$ out of $80$ .

$x = \frac{- 8 \pm i \cdot \sqrt{16 (5)}}{2 \cdot 4}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 8 \pm i \cdot \sqrt{4^{2} \cdot 5}}{2 \cdot 4}$

Pull terms out from under the radical.

$x = \frac{- 8 \pm i \cdot (4 \sqrt{5})}{2 \cdot 4}$

Move $4$ to the left of $i$ .

$x = \frac{- 8 \pm 4 i \sqrt{5}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{- 8 \pm 4 i \sqrt{5}}{8}$

Simplify $\frac{- 8 \pm 4 i \sqrt{5}}{8}$ .

$x = \frac{- 2 \pm i \sqrt{5}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{- 2 - i \sqrt{5}}{2}$

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- 1 \cdot 2 - i \sqrt{5}}{2}$

Factor $- 1$ out of $- i \sqrt{5}$ .

$x = \frac{- 1 \cdot 2 - (i \sqrt{5})}{2}$

Factor $- 1$ out of $- 1 (2) - (i \sqrt{5})$ .

$x = \frac{- 1 (2 + i \sqrt{5})}{2}$

Move the negative in front of the fraction.

$x = - \frac{2 + i \sqrt{5}}{2}$

The final answer is the combination of both solutions.

$x = - \frac{2 - i \sqrt{5}}{2}, - \frac{2 + i \sqrt{5}}{2}$

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

x-intercept(s): $None$

Step 3

Find the y-intercepts.

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

$y = 4 {(0)}^{2} + 8 (0) + 9$

Solve the equation.

Remove parentheses.

$y = 4 \cdot 0^{2} + 8 (0) + 9$

Remove parentheses.

$y = 4 {(0)}^{2} + 8 (0) + 9$

Simplify $4 {(0)}^{2} + 8 (0) + 9$ .

Simplify each term.

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

$y = 4 \cdot 0 + 8 (0) + 9$

Multiply $4$ by $0$ .

$y = 0 + 8 (0) + 9$

Multiply $8$ by $0$ .

$y = 0 + 0 + 9$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 9$

Add $0$ and $9$ .

$y = 9$

y-intercept(s) in point form.

y-intercept(s): $(0, 9)$

Step 4

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 9)$

Step 5

Find the x and y Intercepts 4x^2+8x+9

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