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

Mar 13, 2018

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

#### Explanation:

$25 = {5}^{2}$

Given that, ${x}^{2} - 10 x + 25$

$= {x}^{2} - 10 x + {5}^{2}$

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

Here, $a = x \mathmr{and} b = 5$

$\therefore$ color(magenta)(=(x-5)^2

Mar 13, 2018

It is a perfect square! The square is ${\left(x - 5\right)}^{2}$

#### Explanation:

In a perfect square trinomial, the function ${\left(x + a\right)}^{2}$ expands to:

${x}^{2} + 2 a x + {a}^{2}$

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. $2 a = - 10$

Solving the first equation:

$a = \sqrt{25} \Rightarrow a = \pm 5$

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:

$a = - \frac{10}{2} \Rightarrow a = - 5$

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:

${\left(x + \left(- 5\right)\right)}^{2}$ or ${\left(x - 5\right)}^{2}$

Mar 13, 2018

${x}^{2} - 10 x + 25 = \left(x - 5\right) \left(x - 5\right) = {\left(x - 5\right)}^{2}$

#### Explanation:

A quadratic can be written as $a {x}^{2} + b x + c$

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

• $a = 1$

• is ${\left(\frac{b}{c}\right)}^{2} = c$?

In a perfect square trinomial, a special relationship exists between $b \mathmr{and} c$

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

Consider:
${x}^{2} \textcolor{b l u e}{+ 8} x + 16 \text{ } \leftarrow {\left(\textcolor{b l u e}{8} \div 2\right)}^{2} = {4}^{2} = 16$

${x}^{2} - 20 x + 100 \text{ } \leftarrow {\left(- 20 \div 2\right)}^{2} = 100$

${x}^{2} + 14 x + 49 \text{ } \leftarrow {\left(14 \div 2\right)}^{2} = 49$

In this case:

${x}^{2} - 10 x + 25 \text{ } \leftarrow {\left(- 10 \div 2\right)}^{2} = {\left(- 5\right)}^{2} = 25$

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

${x}^{2} - 10 x + 25 = \left(x - 5\right) \left(x - 5\right) = {\left(x - 5\right)}^{2}$