Is #x^2 − 10x + 25# a perfect square trinomial and how do you factor it?

3 Answers
Mar 13, 2018

Answer:

#color(magenta)(=(x-5)^2#

Explanation:

#25=5^2#

Given that, #x^2-10x+25 #

#=x^2-10x+5^2#

Identity: #color(red)( a^2-2(ab)+b^2=(a-b)^2#

Here, #a=x and b=5#

#therefore# #color(magenta)(=(x-5)^2#

Mar 13, 2018

Answer:

It is a perfect square! The square is #(x-5)^2#

Explanation:

In a perfect square trinomial, the function #(x+a)^2# expands to:

#x^2+2ax+a^2#

If we try to fit the problem statement into this format, we would have to figure out what value #a# is that gives us:

  1. #a^2=25#
  2. #2a=-10#

Solving the first equation:

#a=sqrt(25) rArr a=+-5#

There are two solutions for a there because the square of either a negative or positive real number is always positive.

Let's look at possible solutions for the second equation:

#a=-10/2 rArr a=-5#

This agrees with one of the solutions for the first equation, meaning that we have a match! #a=-5#

We can now write out the perfect square as:

#(x+(-5))^2# or #(x-5)^2#

Mar 13, 2018

Answer:

#x^2-10x+25= (x-5)(x-5) = (x-5)^2#

Explanation:

A quadratic can be written as #ax^2+bx +c#

There is a quick way to check whether it is a perfect square trinomial.

  • #a =1#

  • is #(b/c)^2 = c#?

In a perfect square trinomial, a special relationship exists between #b and c#

Half of #b#, squared will be equal to #c#.

Consider:
#x^2 color(blue)(+8)x +16" "larr (color(blue)(8)div2)^2 = 4^2 =16#

#x^2 -20x+100" "larr (-20div2)^2 = 100#

#x^2 +14x+49" "larr (14 div2)^2 =49#

In this case:

#x^2-10x+25 " "larr (-10div2)^2 = (-5)^2=25#

The relationship exists, so this is a perfect square trinomial.

#x^2-10x+25= (x-5)(x-5) = (x-5)^2#