How do you factor the trinomial #x^2 - 14x + 24#?

1 Answer
Rishi A. · Sidharth
Feb 27, 2016

The trinomial #x^2-14x+24# can be factored as #(x-2)(x-12)#.

Explanation:

Approach 1
Since both #-14# and #24# are relatively small and easy numbers. We can just look for two numbers #a,b# that satisfy the following two equations:

#ab=24#
#a+b=-(-14)=14#.

After some experimentation, we arrive at our answer.

Approach 2
For quadratic equations in general, it often makes sense to simply go straight to the quadratic formula to solve for #x# directly.

It gives you #x=7+-5#. It is trivial to figure out the factorization from here.

Approach 3

We use

#alpha + beta = -b/a#
#alpha * beta = c/a#

In general we can express an equation as

#(x-alpha)(x-beta)#

so
# 14 = alpha + beta#
# 24 = alphabeta#

Now you can solve these 2 equation for # alpha# and # beta#

But on close inspection we find

#alpha = 12, beta = 2#

So the above equation is

#(x-12 )(x-2)#

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