Bob T. answered • 11/11/14
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For the expression 14x^2 + 15x - 9, we have the numbers
-126
21 -6 (21)(-6) = -126; 21 + (-6) = 15
15
Please copy this pattern on paper.
Then, place an x between the four numbers so that the -126 and 15 are above and below, and the 21 and -6 are on the sides. The order of the numbers 21 and -6 can appear as 6 and -21. However, the product a ⋅ c be above and b must appear below. This is the construction of the "magic X." a, b, and c belong to the general quadratic expression ax^2 + bx + c.
The question is bound to arise: How does one produce the 21 and -6? The question you want to ask is: "What two numbers multiplied and added equal to the product (-126, above) and the sum (15, below)?"
Once you have those two numbers (21 and -6), we are ready to determine the GCF of one pair of factors at a time--two at a time! Write the given terms 14x^2 and -9, spaced apart to fit two terms. Next, place the 21x and the -6x between the given terms so that we have
14x^2 [] [] - 9
Can you place the terms in-between? Strategy is very important: The -9 cannot go with the 14 because these two have nothing in common: 2 x ? = 9 --- 7 x ? = 9 --- 3 x ? = 14 --- 9 x ? = 14
However, what about connecting 14x^2 + 21x? with the GCD, 7x? This factor has a relationship with both 14x^2 and 21x.
Do you see how the GCD, 3, relates with the 6 and -9. Wouldn't -3 also work?
Therefore, we have (14x^2 + 21x) (+) (-6x - 9)
Please, keep this middlemost operator a plus--always! From there, it should be easier to factor (-3x - 9) the terms from the inside to give (-3)(2x+3).
Together, we have 7x (2x + 3) + (-3) (2x+3)
Set this to 7x (2x + 3) - 3 (2x + 3)
Notice that (2x+3) is the common factor; pick up the other two terms and place them into a factor.
Therefore, 14x^2 + 15x - 9 = (7x -3)(2x + 3).
Try factoring 2x^2 - x - 1, and try factoring x^2 - x - 2. What do you notice about the product and sum of each of these?
