How do you find the roots for x^2 – 14x – 32 = 0?

2 Answers
Jun 6, 2015

In an equation of the following form

ax^2+bx+c=0

the method to find the roots is:

1) calculate Delta = b^2-4ac
2) if Delta=0 there is only one root x_0=(-b)/(2a)
3) if Delta>0 there are two roots x_(-)= (-b-sqrt(Delta))/(2a)
and x_(+) = (-b+sqrt(Delta))/(2a)
4) if Delta<0 there is no real solution

Example:

x^2-14x-32=0

rarr a=1; b=-14; c=-32

rarr Delta = (-14)^2 - 4 * 1 * (-32) = 196 +128 = 324

Delta>0 therefore we have two roots:

x_(-) = (14-sqrt324)/2 = (14-18)/2 = -4/2 = -2

x_(+) = (14+sqrt324)/2 = (14+18)/2 = 32/2 = 16

Let us check the validity of our results:

(-2)^2-14*(-2)-32 = 4+28-32 = 0 rarr OK

(16)^2-14*(16)-32 = 256-224-32 = 0 rarr OK

Jun 6, 2015

There are several methods we could use. Here's one.

Notice that 2*16=32 and the difference between 2 and 16 is 14.

So, if the signs work out, we can factor.

x^2-14x-32=(x+2)(x-16)

So, x^2-14x-32=0 if and only if

(x+2)(x-16)=0

Thus, we need

x+2=0 or x-16=0

The solutions are:

x=-2, x=16.