There are three methods to factor a quadratic polynomial: Factoring by guessing, "completing the square", and the quadratic formula. On this page you will learn the first method. The next two pages are devoted to the other methods.

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The Quadratic Equation.

Consider the polynomial . If it is reducible, the answer will look like this:

the catch being that we do not know the roots -*a* and -*b*. One popular trick in Mathematics is to work backwards:
Using the FOIL method we can write

From this we see that the two unknowns *a* and *b* have to satisfy

In English: we need two numbers whose product is 3 and whose sum is -4. At this point, start guessing using integer values for *a* and *b*.
There are really only two choices which come to mind: Among integers only the pair 1 and 3 and the pair -1 and -3 have product 3. Checking their sums we see that we have to have *a*=-1 and *b*=-3 (or vice versa). We are done:

We want to find integers *a* and *b* whose sum is 7 and whose product is -8. There are four choices for two integers with product -8:
*a*=1, *b*=-8; *a*=-1, *b*=8; *a*=2, *b*=-4 and *a*=-2 and *b*=4.
Among these, only *a*=-1, *b*=+8 has the required sum 7. We are done:

Did you already gain some practice? Even though the roots are rational numbers instead of integers, I hope the solution will "jump at you" after staring at the polynomial for a few seconds: *a*=1 and *b*=1/2 will work:

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

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