Question

How do you solve `2a ^ { 2} + 24a = - 92`?

Answer 1

No real number solution.

Using imaginary numbers: `x=-6 \pm i sqrt (10)`

Explanation:

`2a^2+24a=-92`
Add 92 to each side
`2a^2+24a+92=0`
Divide both sides by 2
`a^2+12a+46=0`
Apply the quadratic formula, `x=(-b \pm sqrt(b^2-4ac))/(2a)`, where a=1, b=12, and c=46
`x=(-12 \pm sqrt(12^2-4(1)(46)))/(2(1))`
`x=(-12 \pm sqrt(144-184))/(2)`
`x=(-12 \pm sqrt(-40))/(2)`
Because this requires a square root of a negative number, we see that there is no real number solution for this equation.
Using imaginary numbers:
`x=-6 \pm sqrt (-10)`
`x=-6 \pm i sqrt (10)`

Looking at the graph: the equation never reaches -92, meaning there is no solution.