Question

What should be added to 49x2 –
74xy+25y2 to make it a perfect square trinomial?
5
​

Answer 1

Answer:

Perfect-square trinomials are of the form:

a2x2 ± 2axb + b2

...and are expressed in squared-binomial form as:

(ax ± b)2

How does this look, in practice?

Is x2 + 10x + 25 a perfect square trinomial? If so, write the trinomial as the square of a binomial.

Well, the first term, x2, is the square of x. The third term, 25, is the square of 5. Multiplying these two, I get 5x.

Multiplying this expression by 2, I get 10x. This is what I'm needing to match, in order for the quadratic to fit the pattern of a perfect-square trinomial. Looking at the original quadratic they gave me, I see that the middle term is 10x, which is what I needed. So this is indeed a perfect-square trinomial:

(x)2 + 2(x)(5) + (5)2

But what was the original binomial that they'd squared?

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I know that the first term in the original binomial will be the first square root I found, which was x. The second term will be the second square root I found, which was 5. Looking back at the original quadratic, I see that the sign on the middle term was a "plus". This means that I'll have a "plus" sign between the x and the 5. Then this quadratic is:

a perfect square, with

x2 + 10x + 25 = (x + 5)2

Write 16x2 – 48x + 36 as a squared binomial.

The first term, 16x2, is the square of 4x, and the last term, 36, is the square of 6.

(4x)2 – 48x + 62

Actually, since the middle term has a "minus" sign, the 36 will need to be the square of –6 if the pattern is going to work. Just to be sure, I'll make sure that the middle term matches the pattern:

(4x)(–6)(2) = –48x

It's a match to the original quadratic they gave me, so that quadratic fits the pattern of being a perfect square:

(4x)2 + (2)(4x)(–6) + (–6)2

I'll plug the 4x and the –6 into the pattern to get the original squared-binomial form:

16x2 – 48x + 36 = (4x – 6)2

Is 4x2 – 25x + 36 a perfect square trinomial?

The first term, 4x2, is the square of 2x, and the last term, 36, is the square of 6 (or, in this case, –6, if this is a perfect square).

According to the pattern for perfect-square trinomials, the middle term must be:

(2x)(–6)(2) = –24x

However, looking back at the original quadratic, it had a middle term of –25x, and this does not match what the pattern requires. So: