How can this be simplified to the most basic form?

4sec^2x - 11secx - 3

1 Answer
Mar 2, 2018

(secx-3)(4secx+1)

Explanation:

Take a look at the expression. We have a squared term, a first-degree term, and a constant. This can be treated as a quadratic, except instead of dealing with x, we're dealing with secx.

Let's temporarily say u=secx and rewrite everything in terms of u:

4u^2-11u-3

Now this perfectly resembles a quadratic. Let's use the Quadratic Formula to factor:

u=(-b+-sqrt(b^2-4ac))/(2a), a=4,b=-11,c=-3

u=(11+-sqrt(121-(4)(4)(-3)))/(2*4)

u=(11+-sqrt(169))/(2*4)=(11+-13)/8=3, -1/4

u=3, u=-1/4

If u=-1/4, 4u=-1.
We do this because we generally don't want fractions in our factored form when it's avoidable.

Writing in factored form:

(u-3)(4u+1)

We simply moved our solutions for u over to the same side as u, which resulted in switching the signs.

Replacing u with secx:

(secx-3)(4secx+1)