Question

What is a perfect square? Why is + 8x +16" alt="x^{2} + 8x +16" align="absmiddle" class="latex-formula"> a perfect square?

Answer 1

Answer:

It is a perfect square. Explanation below.

Explanation:

Perfect squares are of the form

(

a

+

b

)

2

=

a

2

+

2

a

b

+

b

2

. In polynomials of x, the a-term is always x.(

(

x

+

c

)

2

=

x

2

+

2

c

x

+

c

2

)

x

2

+

8

x

+

16

is the given trinomial. Notice that the first term and the constant are both perfect squares:

x

2

is the square of x and 16 is the square of 4.

So we find that the first and last terms correspond to our expansion. Now we must check if the middle term,

8

x

is of the form

2

c

x

.

The middle term is twice the constant times x, so it is

2

×

4

×

x

=

8

x

.

Okay, we found out that the trinomial is of the form

(

x

+

c

)

2

, where

x

=

x

and

c

=

4

.

Let us rewrite it as

x

2

+

8

x

+

16

=

(

x

+

4

)

2

. Now we can say it is a perfect square, as it is the square of

(

x

+

4

)

.