Question

How do you use completing the square to make a perfect square trinomial from `x^2 - 7/2x` + _?

Answer 1

`49/16`

Explanation:

In order to "complete the square," it's necessary to write a trinomial that is the equivalent to a binomial squared. Examples of perfect binomial squares would include `x^2-6x+9`, which is equivalent to `(x-3)^2`, and `36x^2-12x+1`, which is equivalent to `(6x-1)^2`.

Consider that `(x+a)^2=x^2+2ax+a^2`.
If, for example, we wanted to complete the square with `x^2+10x+?` we know that `10x`, the middle term, must be equal to `2ax`.
Therefore, if `10x=2ax`, we can divide by `2x` on both sides to find that `5=a`.
If we plug this back into the original form of `x^2+2ax+a^2`, we can complete the square by adding `a^2` to the end, which is `25`.

In your case, we can do the same thing. Your middle term, which must be equal to `2ax`, is `-7/(2)x`.

Thus, if `2ax=-7/(2)x`, if we divide both sides by `2x`, we see that `a=-7/4`.

Since `a=-7/4`, when we plug everything back into the form of `x^2+2ax+a^2`, we get `x^2-7/(2)x+color(red)(49/16)`.