M&N Method for Factoring Trinomials

Baldwin Lathrop High School

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1] Make a diamond. The top box gets a times c, and the bottom gets b.

2] Find two numbers, m & n, such that .

3] Rewrite the trinomial as .

Example: , , .

4] Factor each side separately. .

5] If the contents of the two parentheses are equal, then recombine.

6] To completely factor, you can factor out each of the coefficients, in this case, 3 and 2. The 3 and the 2 multiply to 6, leaving us with

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Sample Problems

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Completing the Square

We’ll do this by example. Let’s use #3 from above

3]

First factor out the leading terms coefficient.

Then spread space between the 2^nd and 3^rd terms.

Then do four things to the 2^nd term’s coefficient:

1. Cut it in half.
2. Square it.
3. Add it.
4. Subtract it.

Half of 2 is 1, and 1 squared is 1, so

The first three terms are a perfect square trinomial, which gives us the square of the first variable, then the following algebraic sign, then half the coefficient of the second term. The rest is just junk to deal with in a minute.

Then simplify by distributing the number in front, in this case a 4.

This is AHK Form, or Vertex Form.

To find the x-intercept zeros, set y = 0.

Now solve for x.

This gives us either zero, one or two answers. For this problem we get two answers,

Plugging them back into the original equation will check for accuracy.

Both check!

Quadratic Formula

The quadratic formula is . Let’s try it to make sure we still get the same thing.

3] , and a = 4, b = 8 and c = 3.

The x-intercepts will be at

the same answers.