Quandaries and Queries


Dear Sirs,
I am trying to help out my 9th grader with math problems that she's been given the questions and answers to, but it wasn't explained to her how to come up with the answers. I apologize, but don't know the mathematical terminology for the questions.

Q1: a2 + 5a + 6
Teacher said to write: b * c = 6 AND b + c = 5
Teacher's steps:
_a2 + 3a_ + _2a +6_
= a(a+3) + 2(a+3)
= (a + 2) (a + 3)

I didn't really understand the purpose of the teacher's steps, but was able to get my daughter to find factors of 6 and then add them together to find which resulted in the sum of 5.

Unfortunately, this problem-solving technique did not help us to solve for the more complex problems as follows:

Q2: 9a3 - 24a2 + 12a

Teacher's answer: 3a (3a - 2) (a - 2)

The b's and c's above were added by me. The teacher just left spaces to be filled in. The underlines to the left and write of part of the equation are there because the teacher said to underline each side of the equation, but didn't give the steps for the more complex problem.

Thanks in advance for your quick and detailed response. Your website is WONDERFUL!!




Hi Rise,

For the first problem I would do what you did "find factors of 6 and then add them together to find which resulted in the sum of 5".

For the second problem

9a3 - 24a2 + 12a

the first thing that I see is that there is an a in each term and that 3 divides each term. Hence there is a common factor of 3a. thus

9a3 - 24a2 + 12a
=3a(3a2 - 8a + 4)

You are now back to factoring a quadratic similar to problem 1

3a2 - 8a + 4

First find the factors of 3, the coefficient of a2. The only factors of 3 are 3 and 1. Thus

3a2 - 8a + 4
=(3a      )(a      )

and I only need to find the two missing terms.

Next I look at the constant term in the expression 3a2 - 8a + 4. The constant term is +4 and it's the + that I see first. That tells me that the missing terms have the same sign. Then I look at the middle term, -8a. Since this term is negative the two missing terms must be negative. Thus I have

3a2 - 8a + 4
=(3a -    )(a -    )

Finally find the factors of 4, the constant term. 4 is 4 times 1 or 2 times 2 and hence I have3 possibilities for the two missing terms

(3a -  4)(a -  1)
(3a -  1)(a -  4) and
(3a -  2)(a -  2)

At this point you need to expand each of these possibilities. They each give 3a2 as the first term and +4 as the constant term, so you only have to check the middle term in each case. Only the third option give a middle term of -8a and hence

9a3 - 24a2 + 12a = 3a(3a -  2)(a -  2)



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