Most students have that one thing that they fear in algebra. Some people fear graphing. Some fear exponential equations. Some fear the quadratic formula.
The quadratic formula is a method of factoring used to find the roots or zeros of a quadratic equation, ax2+bx+c=0. The quadratic formula looks like this:
I feared the quadratic formula at one point in my math career, but I’ve learned to embrace this tool. I also learned how it makes my math life easier. Let’s look at it and how to best use the quadratic formula.
What is the quadratic formula?
As mentioned above, the quadratic formula is used to factor or solve a quadratic equation. We use the quadratic formula to find the zeros or the roots of an equation.
When a problem asks you to find the x-intercepts on a graph, they ask you to find where the equation’s x values are equal to zero. They are asking you to factor the equation.
Sometimes finding a factoring a quadratic equation is easy. As you do more and more algebra problems, your brain starts to pick up on patterns in the different problems.
For instance, if you look at the equation, x2+2x-35=0, you can immediately (or with a bit of work) a few things.
The constant, 35, can be broken down into 7 and 5. The 7 and 5 can also be subtracted to get the +2 in the middle term. Therefore, the equation breaks down into (x+7)(x-5).
But what about the equation, 2x2+15x+25? I’m sure many can figure out how to break this one down with no problem. This is where the quadratic equation might be helpful, though.
The journey from the quadratic equation to the quadratic formula.
For those curious, let’s break down how the quadratic equation produced the quadratic formula.
To start this journey, we’ll need to use the technique of completing the squares.
Our starting point is with the quadratic equation, ax2+bx+c=0.
Completing the squares requires that our x2 term not have any coefficients, so we must divide all terms in the equation with a.
Once we divide everything by a, we isolate the x terms on one side and move the constant term to the other side of the equation.
We take the x term (b/a), divide it by two, and then square that. To ensure both sides are even, we add that term to the other side where our constant term is.
Let’s start with the left side of the equation. Because we “completed the squares,” we can take that write that side of the equation as the square of two terms.
On the right side, we combine both of those terms. First, we make sure that the denominators are the same to add both terms. Then we add both terms and combine them.
This is what we have so far.
To get rid of the square on the left side, we have to take the square root of both sides. When we do this, the right side has a +/- sign in front of it.
On the left, we subtract b/2a from both sides to isolate the x.
It should look pretty familiar now. We can combine both of those terms since they share a common denominator.
And that’s how we get the quadratic formula from the quadratic equation.
Instances where the quadratic formula is beneficial.
The best and quickest way to factor a quadratic equation is to visually inspect the equation first. Scout out the equation for any common factors or anything that jumps out at you right away.
An equation like the following has obvious patterns that can be seen immediately. Once you start to see the patterns regularly, your brain realizes what can and cannot be easily factored.
But what about the following?
There is nothing obvious that jumps out in the first equation. The second equation is a mess. Trying to come up with factors through trial and error could take a while.
These instances are where the quadratic formula comes in handy.
Example problems using the quadratic formula.
We’ll tackle the two that I mentioned above for our example problems. One is a bit easier, and the other has pretty large coefficients.
The first equation is 2x2+15x+25=0. I feel that the first one could be factored. Since there are few possible factors for 2 and 25, we could easily factor this equation on our own. But let’s try it with the quadratic formula instead, just for practice.
Our coefficients for the quadratic equation are a=2, b=15, and c=25. If those coefficients are negative expressions, the negative sign would be carried into the quadratic formula.
Here is what our quadratic formula looks like when we substitute our values in.
Once we work through the calculations, we find that the factors or roots of this equation are x=-5 and x=-10/4.
The following example is a little trickier to figure out. Maybe it could be factored with a bit of work, but I find that when the coefficients are large, it’s easier to use the quadratic formula.
Let’s take a look at the equation 125x2+235-30=0.
Our first step is to pull out all the coefficients from our quadratic equation.
The next step is to start plugging the coefficients into the quadratic formula. Once we have them all in, we can perform the calculations.
Most of the tools and formulas found in math aren’t there to intimidate you. Once you learn how to use them, they help make your math career a lot easier.
Stay consistent with your homework and keep practicing!