Question 366759: please help me find the real solution of this equation by factoring Found 2 solutions by mananth, jsmallt9:Answer by mananth(16946) (Show Source):
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To solve a 3rd degree equation like this we want one side of the equation to be zero and then factor the other side. We already have one side zero so we can go straight to the factoring.
The Greatest Common Factor (GCF) is 1 (which we almost never bother factoring out). Moving to other factoring techniques, the expression on the left has 4 terms. This is too many for the five most commonly taught factoring patterns and it is too many terms for trinomial factoring. So all we have left if factoring by grouping or factoring by trial and error of possible rational roots. Since the last one is probably the hardest way to factor, we'll try factoring by grouping first.
When I factor by grouping I find it easier if the expression is written with additions. So I'm going to use:
When factoring by grouping you look for subexpressions (or "groups") that have a GCF. With your equation the first two terms have a GCF of and the last two terms have a GCF of 1 (or -1). (Note: Factoring by grouping is one of those times where we might bother factoring out a 1!) Factoring these out we get:
Now, if we're lucky, the "non-GCF" factors will match. Our "non-GCF" factors are x-3 and -x+3 which are not the same. But they are negatives of each other. So if we factor out -1 instead of 1 from the second group we will end up with matching "non-GCF" factors:
Now that we have matching "non-GCF" factors in each group, we have a GCF between the two groups: x-3. We can factor this out from the two groups:
or
In one sense, factoring is like redcuing fractions. You keep going until you can't go any further. In the case, our second factor, is a difference of squares which can be factored by that pattern. So now we have:
We're done factoring. From this we can find the solutions. The Zero Product property tells us that this (or any) product can be zero only if one of the factors is zero. So:
x-3 = 0 or x-1 = 0 or x+1 = 0
Solving these we get:
x = 3 or x = 1 or x = -1
These are the 3 real solutions for your equation.
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