# How do you solve the quadratic 7n^2+6=-40 using any method?

May 3, 2018

$x = \frac{\pm i \sqrt{322}}{7}$

#### Explanation:

$7 {n}^{2} + 6 = - 40$

To use the quadratic formula to find the zeroes, we need to make sure the equation is written in the form $\textcolor{red}{a} {x}^{2} + \textcolor{m a \ge n t a}{b} x + \textcolor{b l u e}{c} = 0$.

To do so, we need to add $\textcolor{\mathmr{and} a n \ge}{40}$ to both sides of the equation:
$7 {n}^{2} + 6 \quad \textcolor{\mathmr{and} a n \ge}{+ \quad 40} = - 40 \quad \textcolor{\mathmr{and} a n \ge}{+ \quad 40}$

$7 {n}^{2} + 46 = 0$

So we know that:
$\textcolor{red}{a = 7}$

$\textcolor{m a \ge n t a}{b = 0}$

$\textcolor{b l u e}{c = 46}$

The quadratic formula is $x = \frac{- \textcolor{m a \ge n t a}{b} \pm \sqrt{{\textcolor{m a \ge n t a}{b}}^{2} - 4 \textcolor{red}{a} \textcolor{b l u e}{c}}}{2 \textcolor{red}{a}}$.

Now we can plug in the values for $\textcolor{red}{a}$, $\textcolor{m a \ge n t a}{b}$, and $\textcolor{b l u e}{c}$ into the quadratic formula:

$x = \frac{- \textcolor{m a \ge n t a}{0} \pm \sqrt{{\left(\textcolor{m a \ge n t a}{0}\right)}^{2} - 4 \left(\textcolor{red}{7}\right) \left(\textcolor{b l u e}{46}\right)}}{2 \left(\textcolor{red}{7}\right)}$

Simplify:
$x = \frac{\pm \sqrt{- 1288}}{14}$

The solution is imaginary since we cannot take the square root of a negative number.

However, we know that $i$, or imaginary, is equal to $\sqrt{- 1}$. Therefore, we can factor out a $- 1$ from the square root and make it an $i$:

$x = \frac{\pm i \sqrt{1288}}{14}$

$x = \frac{\pm 2 i \sqrt{322}}{14}$

$x = \frac{\pm i \sqrt{322}}{7}$

This is the same thing as:
$x = \frac{i \sqrt{322}}{7}$ and $x = \frac{- i \sqrt{322}}{7}$
because $\pm$ means "plus or minus."

To clarify, this doesn't mean that there is a zero or root. This is an imaginary solution, meaning that there are no zeros. To prove this, let's look at the graph of this equation:

(desmos.com)

As you can see, there are no zeroes. The vertex starts above the $x$-axis.

Hope this helps!