Jon P. answered • 03/25/15
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Knowledgeable Math, Science, SAT, ACT tutor - Harvard honors grad
Where you went wrong was:
x = -b + √(b^2 - 4ac) / 2a
x = 1 + √(1^2 - 4*1*110) / 2
x = 1 + √(1^2 - 4*1*110) / 2
Once you did that, you also made at mistake at:
x = 1 + √-439 / 2
x = 1 + 20.95 / 2
x = 1 + 20.95 / 2
√-439 is an "imaginary number". If you take the square root of a negative number, the answer is not a real number. It would be approximately 20.95i, not 20.95.
Anyway, you can solve this much more easily by factoring:
x2 - x - 110 = 0
factors to:
(x - 11)(x + 10) = 0
So x = 11, or x = -10.
Lynn L.
Also, how do you choose which numbers to use when you are factoring? You could have used 55 and 2 or 5 and 22, etc. i know those numbers wouldn't have worked out but how do you know that without plugging every one of them into the equation?
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03/25/15
Jon P.
tutor
Factoring is a major subject by itself. It sounds like you may not have had any experience with it yet, and it's hard for me to explain the whole thing in a short write-up.
I can explain this particular one though.
When you multiply two binomials in parentheses by each other, you multiply each term in the first one by each term in the second one, and add up the results. So for (x - 11)(x + 10), you would do the following:
x * x = x2
x * 10 = 10x
-11 * x = -11x
-11 * 10 = -110
You can see that there's an x2 and a -110, as in the original quadratic expression. You can also see that if you add 10x + -11x, you get -1x = -x. So that's the correct number of x's for the original expression.
You're right that sometimes you do have to plug in every possibility in order to figure it out. But very often it's easier than that.
In this case you see a couple of things:
- The constant term is -110. That means that the two constants terms in the factors will multiply together to 110, and that one will be positive and one will be negative.
- The coefficient of x is -1. That means that the two constant terms in the facts will be different by 1. So you'd look for two numbers that multiplied together to 110, but were different by just 1. 11 and 10 are just right.
- Since the coefficient of x is negative, it means that the sign of the larger constant in the factors will be negative. And the other sign will be positive. So it will be -11 and +10. That gives you -110 as the constant, and -x for the x term.
- All of this depends on the coefficient of x2 being 1. If it isn't, it gets more complicated.
I hope that helps as a start, but I know it's not at all a complete guide about how to factor.
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03/26/15
Lynn L.
03/25/15