# How do you find the zeros, if any, of y= -4x^2 - 8 using the quadratic formula?

May 13, 2016

Since $y$ is less than zero for all Real values of $x$ there are no Real zeros.
However the quadratic formula can be used to determine the Complex zeros: $x = \pm \sqrt{2} i$

#### Explanation:

The quadratic formula tells us that for a quadratic in the form:
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{a} {x}^{2} + \textcolor{b l u e}{b} + \textcolor{g r e e n}{c} = 0$
the zeros occur at
$\textcolor{w h i t e}{\text{XXX}} x = \frac{- \textcolor{b l u e}{b} \pm \sqrt{{\textcolor{b l u e}{b}}^{2} - 4 \textcolor{red}{a} \textcolor{g r e e n}{c}}}{2 \textcolor{red}{a}}$

Converting the given
$\textcolor{w h i t e}{\text{XXX}} y = - 4 {x}^{2} - 8$
into the required form:
$\textcolor{w h i t e}{\text{XXX")y=color(red)(""(-4))x^2+color(blue)(0)x+color(green)(} \left(- 8\right)}$

The zeros occur at
$\textcolor{w h i t e}{\text{XXX}} x = \frac{\textcolor{b l u e}{0} \pm \sqrt{{\textcolor{b l u e}{0}}^{2} - 4 \left(\textcolor{red}{- 4}\right) \left(\textcolor{g r e e n}{- 8}\right)}}{2 \left(\textcolor{red}{- 4}\right)}$

$\textcolor{w h i t e}{\text{XxXX}} = \pm \sqrt{- 2}$

$\textcolor{w h i t e}{\text{XxXX}} = \pm \sqrt{2} i$

May 13, 2016

There is no 'Real Number' solution for $y = 0$ so the graph does not cross the x-axis.

However there is a solution for $y = 0$ within the set of numbers called Complex Numbers, and that is $x = \pm i \sqrt{2}$

The solution for $x = 0$ is $y = - 8$

#### Explanation:

The x-intercepts occur at the points where the graph crosses the x-axis. This is when y=0.

Given:$\text{ } y = - 4 {x}^{2} - 8$

Substitute 0 for y giving:

$\text{ } 0 = - 4 {x}^{2} - 8$

Multiply both sides by $\left(- 1\right)$ giving

$\text{ } 0 = 4 {x}^{2} + 8$

Subtract 8 from both sides

$\text{ } 0 - 8 = 4 {x}^{2} + 8 - 8$

But $8 - 8 = 0$

$\text{ } - 8 = 4 {x}^{2}$

Divide both sides by 4

$- \frac{8}{4} = \frac{4}{4} \times {x}^{2}$

But $\frac{4}{4} = 1$

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

Take the square root of each side

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

There is no 'Real Number' solution to $x$ so the graph does not cross the x-axis.

However there is a solution within the set of numbers called complex numbers and that is $x = \pm i \sqrt{2}$