This method of factoring trinomials was shown to me

. . by one of my students many years ago.

You may find it as fascinating as I did.

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Example:

Write the two pairs of parentheses and "split the x's": .

Use the first coefficient: .twice

Multiply the first coefficient by the last coefficient: .

We will factor into two parts.

Note the sign of the last term.

. . If "+", thinksum.

. . If "-", thinkdifference.

Since the last sign is "+", factor into two parts

. . whosesumis the middle coefficient .

To factor into all possible pairs, divide by 1, 2, 3, ...

. . keeping those that "come out even".

The pair with a sum of 35 is: and

Since the middle coefficient is , we will use: and

Insert them into the parentheses: .

Factor out all common factors and: .discard them

Answer: .

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Example: .

We have: .

Then: .

The last term is negative, thinkdifference.

Factor into two factors whose difference is

The pair with a difference of 11 is and

Since we want , we will use: and

Insert them into the parentheses: .

Factor out common factors and discard them: .

Answer: .

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At this point, someone will say, "Why not use the Quadratic Formula?"

. . Why not, indeed?

In fact, why bother teaching Factoring at all?

. . And I don't have an answer to that question.

But it would be a shame to run through the Quadratic Formula

. . to factor, say, or