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Module 21.2
Solving EquationsBy Factoring πππ + ππ + π
How can you use factoring to solve quadratic equations in standard form for which a β 1?
P. 997
Until now weβve been factoring quadratic expressions where the leading coefficient βaβ has been 1.
For example:
What do we do when the leading coefficient is NOT equal 1?
For example:
We canβt use the standard form of π₯ + ? π₯ + ? because of the π.
There are a few different methods that are used to factor these, such as Slide & Divide, Guess & Check, Tic-Tac-Toe, and Grouping.Weβre going to learn the Box method.
ππ + ππ + π
πππ + πππ + π
πππ + πππ + π
BOX METHOD - 9 Steps
1) Determine if thereβs a GCF for all 3 terms. If yes, then factor it out.Here, the GCF is 2, so it becomes:
2) Before: Find all factors that multiply to c and add up to bNow: Find all factors that multiply to aΒ·c and add up to b.Here: Find all factors that multiply to 6 and add up to 5.
1,62,3 <<== Which gives us 2x and 3x
π(πππ + ππ + π)
a b c
3) Create a 2x2 table.In the upper left, put the first term.In the lower right, put the last term.In the other two, put the 2 new terms from the previous step. (It doesnβt mater which of those 2 goes where.)
a b c
πππ ππ
ππ π
π(πππ + ππ + π)
4) Between the top 2 boxes, determine the GCF.Write that to the left.
2x πππ ππ
ππ π
5) Divide the upper-left box by the number you just wrote,and write that new number on top of the upper-left box.
2x πππ ππ
ππ π
x
π(πππ + ππ + π)
6) Divide the upper-right box by the number you wrote to the left,and write that new number on top of the upper-right box.
2x πππ ππ
ππ π
x 1
7) Divide the lower-left box by the number you wrote at the top,and write that new number to the left of the lower-left box.You should now have what looks like a multiplication table.
2x3
πππ ππ
ππ π
x 1
π(πππ + ππ + π)
8) Use the numbers youβve written to create two binomials,and combine it with the GCF from the first step, if any.
2x3
πππ ππ
ππ π
x 1
π π + π ππ + π
9) Check your work by multiplying this out (via FOIL).Does it equal the original expression?
π π + π ππ + π = π πππ + ππ + ππ + π
= π πππ + ππ + π
= πππ + πππ + π
Yes!
πππ + πππ + ππ
Letβs try another one.
1) Is there a GCF? No.2) Find all factors that multiply to 60 and add up to 19.
1,602,303,204,15 <<== Which gives us 4x and 15x
3) Create a 2x2 table with the 4 terms.
πππ ππ
πππ ππ
4) Between the top 2 boxes, determine the GCF, and write that to the left.5) Divide the upper-left box by the number you just wrote,
and write that new number on top of the upper-left box.6) Divide the upper-right box by the number you wrote to the left,
and write that new number on top of the upper-right box.7) Divide the lower-left box by the number you wrote at the top,
and write that new number to the left of the lower-left box.8) Use the numbers youβve written to create two binomials,
and combine it with the GCF from the first step, if any.9) Check your work by multiplying this out (via FOIL).
2x5
3x 2
ππ + π ππ + ππππ ππ
πππ ππ
Practice:
πππ β πππ β ππ
πππ β ππ + π
P. 999-1000
βπππ + ππ + π
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