Question

Is `9x^2 + 42x + 49` a perfect square trinomial and how do you factor it?

Answer 1

See below

Explanation:

We know that `(a+b)^2=a^2+2ab+b^2`

In our case: `9x^2+42x+49=3^2·x^2+2·9·7·x+7^2=(3x+7)^2` This it's the same that

`(3x+7)(3x+7)` is the factorization

Answer 2

If `9x^2 + 42x + 49` is a perfect square, then it must fit the pattern:

`(a+b)^2 = a^2+2ab+b^2`

To verify whether it fits the pattern, we set the first term of the pattern equal to the first term of the given trinomial:

`a^2=9x^2`

Then we set the last term of the pattern equal to the last term of the given trinomial:

`b^2=49`

Taking the square root of both we obtain:

`a = 3x` and `b =7`

We can use this information to check whether the middle term is true:

`2ab = 42x`

`2(3x)7 = 42x`

`42x = 42x`

The middle term is true, therefore, we conclude that the given trinomial is a perfect square.

We can use the left side of the pattern to factor the given trinomial:

`(3x+7)^2 = 9x^2 + 42x + 49`