# How do you solve the quadratic 8n^2-5=-110 using any method?

Dec 29, 2016

No Real solutions are possible for the given equation.
Complex solutions: $\pm i \sqrt{\frac{105}{8}}$

#### Explanation:

Real Solutions
$8 {n}^{2} - 5 = - 110 \textcolor{w h i t e}{\text{XX")rarrcolor(white)("XX}} 8 {n}^{2} = - 105$
but $8 > 0$ and ${n}^{2} \ge 0$ for all Real values of $n$
so $8 {n}^{2} \ge 0$ and can not be equal to a value less than $0$

Complex Solutions
$8 {n}^{2} = - 105 \textcolor{w h i t e}{\text{XX")rarrcolor(white)("XX}} {n}^{2} = - \frac{105}{8}$

${n}^{2} = \frac{105}{8} \cdot {i}^{2} \textcolor{w h i t e}{\text{XXX")rarrcolor(white)("XXX}} n = \pm \left(\sqrt{\frac{105}{8}}\right) i$
...alternately, you could apply the quadratic formula to $8 {n}^{2} + 0 n + 105 = 0$ (and get the same result).