Factoring trinomials: 2a^2 + 3a + 1

[sundaybest]

So, my problem is 2a^2 + 3a + 1. What I thought I had to do to factor this, would be to figure out what multiplies to equal 2 (2 * 1), and adds to 3. So, I figured it would be 2 and 1, making the answer (2a + 2)(a + 1). But when I checked it, this doesn't work. When I looked in the back of the book to check my answer it was (2a + 1)(a + 1). I'm not sure how I would get this answer because 1 + 1 = 2 and 1 * 1 = 1, which doesn't go along with the method I tried to use originally. Why is this?

[alan]

When the coefficient of the x squared term is not 1 you need to use the method of decomposition to factor the trinomial.

The first part you have done correctly.
Find two numbers whose product is a*c and whose sum is b.
In this case a*c=2 and b=3 so the two numbers are 2 and 1.

Now, since the x squared coefficient is not 1 you must use the method of decomposition to factor the trinomial.

You replace the middle term with the two numbers you found. So 2a^2+3a+1 gets rewritten as 2a^2 +2a + a + 1.
Then group the first two terms and last two terms factoring out the largest factor you can out of each group.
So you have 2a(a+1)+(a+1).
Now take (a+1) out as a common factor giving (a+1)(2a+1)

Here is a link with a very clear example of how this works.
http://mathcentral.uregina.ca/QQ/database/QQ.09.06/h/kim1.html

[jg.allinsymbols]

A somewhat less refined but logical approach is to fill this with possibilities to test:

Code: Select all

(2a + ___)(a + ___)

...but will anything work? Your only choices to make product of 1 are 1 X 1.

[strumbore]

Those are REALLY good suggestions, the last one is actually the fastest way to solve the problem.

But remember that if you ever get stuck, there is a guaranteed solution to any quadratic equation using the Quadratic Formula (and if there isn't, it means the equation is unsolvable, or at least has no real solutions)

x = -b +- SQRT(b² - 4ac]) all-over-2a