SOLUTION: I need to find the missing square roots (?) of the following.. (7+ the sq root of ?)(7- the sq root of ?) = 44 Thanks in advance for your time and your help. Ramona4757@L

Question 391503: I need to find the missing square roots (?) of the following..
(7+ the sq root of ?)(7- the sq root of ?) = 44
Thanks in advance for your time and your help.
Ramona4757@Live.com
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
The right side of your equation, 44, has no square roots in it. The left side is the product of two binomials (two-term expressions). The only way a product of two binomials which have square roots in them can result in an expression without square roots is if the binomials are conjugates.

The general form for conjugates is:
a+b and a-b
We know from our factoring patterns what the product of conjugates will be:

Looking at the right side, we see two terms, each of which is a perfect square. This means that if "a" or "b" have a square root in them, then after multiplying (a+b)(a-b) the square roots will end up being squared. And square roots that are squared will no longer be square roots.

The product of binomials that are not conjugates will end up with 3 or 4 terms and they will not all be squared terms. So if there are square roots in the non-conjugate binomials, there will still be square roots in the product.

So we have determined that the left side of your equation must be a product of conjugates (since the result has no square roots). As we can see, the first terms of the binomials, the "a", is 7. The second terms, the "b", must be the same,too, in order for these to be conjugates. So the ?'s must both be the same number. Let's call this number "x". So the equation must be:

Using the factoring pattern to multiply out the left side we get:

which simplifies to:
49 - x = 44
Subtracting 49 from each side we get:
-x = -5
Dividing (or multiplying) both sides by -1 we get:
x (or ?) = 5

P.S. I hope this also answers your other question about conjugates.
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