Question

what must be added to eachside
of
${x}^{2} + 12x = 64$
to get perfect square?find the value of x
​

Answer 1

Answer: Hope this helps! (Mark me as brainleist plz!)

There is one "special" factoring type that can actually be done using the usual methods for factoring, but, for whatever reason, many texts and instructors make a big deal of treating this case separately. "Perfect square trinomials" are quadratics which are the results of squaring binomials. (Remember that "trinomial" means "three-term polynomial".) For instance;

$(x + 3)2= (x + 3)(x + 3)= x2 + 6x + 9...so x2 + 6x + 9$

is a perfect square trinomial.

Step-by-step explanation:

Recognizing the pattern to perfect squares isn't a make-or-break issue — these are quadratics that you can factor in the usual way — but noticing the pattern can be a time-saver occasionally, which can be helpful on timed tests.

The trick to seeing this pattern is really quite simple: If the first and third terms are squares, figure out what they're squares of. Multiply those things, multiply that product by 2, and then compare your result with the original quadratic's middle term. If you've got a match (ignoring the sign), then you've got a perfect square trinomial. And the original binomial that they'd squared was the sum (or difference) of the square roots of the first and third terms, together with the sign that was on the middle term of the trinomial.

Perfect-square trinomials are of the form:

$a2x2 ± 2axb + b2$

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

$(ax ± b)2$

Example 2.

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?

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

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