Question

How do you solve `a^ { 2} - 18a = - 94`?

Answer 1

Re-write in standard parabolic form and apply the quadratic formula to obtain Complex solution values for `a`

Explanation:

If `a^2-18a=-94`
then (in standard form)
`color(white)("XXX")a^2-18a+94=0`

Since this is a parabola (opening upward)
its minimum occurs when the derivative of the left side `=0`;
that is when `2a-18=0color(white)("xxx")rarrcolor(white)("xxx")a=9`
and the minimum value would be
`color(white)("XXX")9^2-18 * 9 +94`
which is greater than zero; so there are no Real valued solutions.

We can apply the quadratic formula to get solutions at
`color(white)("XXX")a=(18+-sqrt(18^2-4 * 1 * 94))/(2 * 1)`
which simplifies as (assuming I did the arithmetic correctly)
`color(white)("XXX")a=9+-sqrt(42)/2i`