# Blog

## Factoring Quadratic Equations the Easy Way

When I was in high school, I remember having acute anxiety about having to factor quadratic equations on tests and quizzes. This was the ambiguous section of high school algebra when all trinomials were magically factorable into nice, whole integers – the section before learning the ever-useful Quadratic Formula.

The reason I had this anxiety was that it seemed as though there was no precise algorithm for factoring trinomials. You simply had to play with your a, b, and c’s until you found two numbers that multiplied to equal c, and added to equal b. Easy, right? But the problem is that a lack of precision is stressful at that age. So how do you cope?

One of my student’s teachers showed her class how to algorithmically find the factors of a integer factorable trinomial. I’ve formalized it as a theorem for specific trinomials, and I’d love to share it with you today. I’ve also provided the theorem’s quick proof, and perhaps most importantly, the general method for finding factors based on this theorem. This nice little theorem should find its home as a staple in algebra textbooks.

Given a trinomial/quadratic expression of the form

that can be factored into integers, we can find a factorization of the form

such that

and

It follows that, given a trinomial/quadratic equation of the form

that can be factored into integers, we can find roots of the form

and

such that

and

## Method to Teach to Students

Quadratic Expressions: You are given a quadratic expression . Our process for finding factors is as follows.

1. First, multiply and . List all of the factor pairs, positive and negative, that are equivalent to .

2. From the list of factor pairs you determined, find the set that, when the elements are added together, equal . Call this pair . This gives us that

and

3. Once you determine and , rewrite your quadratic expression with and substituted for :

4. Now group the expression so that is couples with and is coupled with . It is important to note that and must share a factor, just as and must share a factor.

5. Pull out a common factor from the elements of the first set of parentheses.

6. Pull out a common factor from the elements of the second set of parentheses. Now the elements in the first and second set of parentheses should be the same.

We can see this by re-writing as , which comes from

Once we do this, we have

Now we have that the quantities in both sets of parentheses are equal. We can now factor out the expression inside the parentheses and get our factors:

To isolate in the second set of parentheses, we factor out the and get

We can see that our factors are and .

Quadratic Equations: Finding roots of a quadratic equation follows the same process, but we take it one step further; we set our factors equal to zero and solve for the two possible solutions.

and

Then

and

are the roots of the quadratic equation.

## Quick Theorem Proof

This actually follows the same algorithm as the Method to Teach to Students. I’ll show it here for quadratic equations.

Given a quadratic equation , that can be factored into integers, such that and , we have that

thus

and

Note: Another interesting investigation would be to see if this works for quadratic expressions that are not factorable into integers.

## Introductory Video for Academic Institutions

Our first blog post introduced you to the TranscendED team and the various services we offer for individual students. Today we’d like to give everyone a sneak peek of the services that we offer to academic institutions! Check it out in the video below.

## Science IRL

This week we’d like to feature Science IRL, one of our favorite YouTube series! Their latest video features TranscendED’s very own Dr. Lisa Rogers, along with a few other fabulous female scientists, discussing various aspects of their careers in science: how their interest in science began, what they do with science, and what advice they have for students interested in a career in science.

Science IRL, created by and starring systems biologist Molly Edwards, is a web series that shows viewers what being a scientist is like In Real Life. In every episode, Molly shows us an experiment involving that month’s topic, such as DNA, polymerase chain reactions (PCR), or mitosis.

Check out all the current videos available, and be sure to keep an eye out for upcoming episodes!