Question

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

Answer 1

`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`

Answer 2

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`

Answer 3

`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`