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To solve this we need to factor it. We start with Greatest Common Factor (GCF) factoring. But the GCF here is 1 and we rarely bother factoring out a 1. Next we look to any of the other factoring techniques.
The left side has four terms. This is too many terms for any of the factoring patterns and too many terms for trinomial factoring. That leaves us with factoring by grouping and factoring by trial and error of the possible rational roots. Since there are many factors of 500, I'm going to try factoring by grouping first.
When I factor by grouping I start by rewriting the polynomial as additions. By having only additions I:
get to use the Commutative and/or Associative Properties which is often needed when factoring by grouping.
avoid the errors that are easy to make when there are subtractions present.
Now we'll start the factoring. You are looking for subexpressions which have a GCF which can be factored out. The first two terms have a GCF of and the last two terms have a GCF of 100 (or -100). Factoring these GCF's out of each subexpression we get:
Now we'll look at the "non-GCF" factors (in red). We hope that they match. These do not match. But they are opposites of each other. So if we factor out a -1 from one of them they will end up matching. (This means that -100 was the better GCF to use with the last two terms.)
Now the non-GCF factors match. We factor the non-GCF factors out of each group:
We've succeeded at factoring by grouping. But factoring is like reducing fractions in the sense that you keeping going until you can't go any further. The (or ) factor is a difference of squares which can be factored using that pattern: . Using this pattern we get:
Since none of these factors will factor any further we are done factoring. Now that the equation is factored we can solve it. The Zero Product Property tells us that this product is zero only if one of the factors is zero. So: or or
Solving these we get: or or
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