Please Help Math problem!!!!!!!!!!!!!?

Question

a.) Explain why solving (x - 4) (x-7) = 6 by setting each factor equal to 6 is not correct.

b.) Determine the correct solution to (x-4) (x-7) = 6

Answer 1

a.) Here's one way of looking at this problem. Let's say that we set one factor, x - 4, equal to 6. Then x - 7 must be equal to 1, since only 6 x 1 = 6. Clearly then x must be equal to 10, because x - 4 = 6 = 10 - 4. Now, since x is fixed at 10, we substitute it into the second factor. Oh oh! We have run into a problem here. 10 - 7 = 3 ╪ 1. So something must be wrong with our method.

As one answerer has indicated, only when we assign a factor's value to equal 0 does our method work, because if x makes the value of that factor equal to 0, then it doesn't matter what the value of the other factor is. 0 times (x - c) = 0 for all values of x and all values of c.

b.) (x-4) (x-7) = 6
x² - 11 x + 28 = 6
x² - 11 x + 22 = 0

This last equation doesn't factor, so use the quadratic formula:

x = [- b ± √(b² - 4ac)] / 2a
x = [11 ± √(121 - 4(1)(22)] / 2(1)
x = [11 ± √(121 - 88)] / 2
x = [11 ± √33] / 2

Answer 2

Well for one thing, try it, it doesn't work. If you set any individual factor to 6, you haven't taken into account what happens when you multiply it by the other factor. Remember 0 is the only number that when multiply it by anything else gives a result of 0.You need to multiply the left side out, then subtract 6 from each side so the right side is equal to 0. Then you can solve the resulting quadratic. I think this technique is called the zero factor theorem.

Answer 3

a) because x-4 isn't necessarily equal to 6 and neither is x-7.
b) use the F.O.I.L. method. (First, Outside, Inside, Last)then combine like terms and solve.
1. multiply x by x, multiply x by -7, multiply -4 by x, multiply -4 by -7. (you get x^2- 11x + 28 = 6)
2. subtract the 6 (x^2 -11x +22=0)
3. then use the quadratic equation. here's a link:
http://www.purplemath.com/modules/quadform.htm

normally I would finish the steps on my own, but I have other stuff..... I hope this helped.

Answer 4

It doesn't work because only 6 times 1 = 6 so unless by setting one factor to be 6, the other actually comes out to be 1 (very unlikely) it won't work.

However, if the answer were zero, then as long as one cactor came out zero it wouldn't matter what the other was because anything times zero is zero.

So multiply out the left, subtract 6 from both sides to make it equal zero, then do FOIL and see what makes each binomial equal zero.