How can this be simplified to the most basic form?

4sec^2x - 11secx - 34sec2x11secx3

1 Answer
Mar 2, 2018

(secx-3)(4secx+1)(secx3)(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,x, we're dealing with secx.secx.

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

4u^2-11u-34u211u3

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=-3u=b±b24ac2a,a=4,b=11,c=3

u=(11+-sqrt(121-(4)(4)(-3)))/(2*4)u=11±121(4)(4)(3)24

u=(11+-sqrt(169))/(2*4)=(11+-13)/8=3, -1/4u=11±16924=11±138=3,14

u=3, u=-1/4u=3,u=14

If u=-1/4, 4u=-1u=14,4u=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)(u3)(4u+1)

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

Replacing uu with secx:secx:

(secx-3)(4secx+1)(secx3)(4secx+1)