Using the Quadratic Formula

So, now that you know the answer to ‘What is the Quadratic Formula,’ next I will show you examples of using it. Refer back to my last post to familiarize yourself with what the quadratic formula looks like. I’ve also explained there the nature of the roots of a quadratic equation. If you haven’t read it, I recommend taking a look as it might help you to visualize and to find the solution to a quadratic equation easier.

For my first example of using the quadratic formula to find the roots of a quadratic equation, let’s keep it simple.

x2 – 2x –3 = 0

Comparing this to the standard form of a quadratic equation, ax2 + bx + c = 0, we can equate the letter coefficients to the values provided. That is, we can say that a = 1, b = (–2), c = (–3). Now, we can simply substitute these values into the quadratic formula:

So, we have:

If you follow along with the arithmetic, you can see that we’ve solved the quadratic formula to show that the roots of the given equation are x = 3 and x = (–1).

Now, remember that I said in a previous lesson that you have to check your answers! Substitute these values back into the original equation, and you will find that they do indeed satisfy the equation. So, these are the correct roots!

Of course, you may have noticed that this question didn’t actually require the quadratic formula to solve for the roots. The quadratic formula worked well and got us the answer, but as you saw, it required a bit of work. And more work means more opportunity to make a mistake! You may have noticed that there was actually a faster way of solving the question. If you noticed that you could reduce the question down to (x – 3)(x + 1) = 0, you could simply let each set of brackets equal zero, and then find again that x = 3 and x = (–1) are the correct solutions.

Let’s try another one, adding some more of the previous math concepts I’ve gone over.

Using the quadratic formula, find the roots of: 2x3 + 3x2 = 4x

It’s looks a little more complicated than the last one, huh? It has higher order exponents, and it doesn’t immediately look like a quadratic equation, as the first example did. However, with a little bit of arithmetic, and using your skills from the math concepts I explained in my post about factoring (specifically, Grouping in my Methods of Factoring post), it will begin to look a bit more familiar and workable.

So then, apply grouping techniques to our question. Let’s bring everything to one side first though. Recall that the standard form of a quadratic equation equals zero.

2x3 + 3x2 = 4x
2x3 + 3x2 – 4x = 0
x(2x2 + 3x – 4) = 0

Looks a little better now, right? Maybe, something that might fit into the quadratic formula? Recall that the roots, or solutions, are any values of x that make the expression true. So, what we have derived up to this point is a product of two expressions that equals zero, and therefore the roots will be whatever values of x cause each part of the product to equal zero. The first (potential) root is obvious, from the first of the two expressions in the product: x = 0. (Substitute 0 back into the original equation to verify this is a correct root!) The second part, 2x2 + 3x – 4, will require more work, and if we let it equal zero, you can see that it will fit into the quadratic formula perfectly.

To prepare for the quadratic formula, we need to identify our a, b, and c values. They are: a = 2, b = 3, and c = (–4). Now, we just substitute into the formula, do the math, and come up with our root(s) for this part of the question!

So, these are our answers for the two roots to the quadratic expression part of our original question. These are the radical forms of the solutions, so they look way more complicated. But, often the quadratic formula doesn’t reduce all the way down to a nice, round number and you will be left with something like this. The last thing you have to do is substitute them back into the original question to verify the roots are true, and that is it! Of course, when you write your answers down, make sure you remember to include the roots from the first part of the question, i.e. the part we created by grouping and solved for x = 0.

That last question goes over a lot of math concepts and is definitely comparable to some of the more complicated math questions you may find in your homework or on exams. Review and study it and make sure you understand it. I’ll post another example as well soon, if anyone needs some more examples of using the quadratic formula.

The Quadratic Formula

Following my posts on How to Solve Quadratic Equations (here and here), you will soon find that not all quadratic equations can be solved by quadratic factoring, and you will come to rely on The Quadratic Formula to help you. As a quick refresher, a quadratic equation is one which takes the form of ax2 + bx + c = 0, as long as the “a” term is not zero. In other words, a quadratic equation is one in which there is an x2. (The “b” or “c” term can be zero.) I have already described the process you should follow if your question can be factored down, and you can express it as a product of two smaller expressions. Then, you can solve for two roots by letting each of the small expressions equal zero. I highly recommend reading my previous post if you need to go over this quadratic factoring technique.

However, as I said, not all quadratic equations can be solved this way. Sometimes, they are already expressed in a simplest form, or further manipulations just make things messier. In these cases, you can use The Quadratic Formula to solve for the roots of the equations. At first glace, the quadratic formula looks like a beast of a formula to use, and even harder to memorize! But, trust me… commit this formula to memory and learn how to use it, and solving quadratic equations will become so easy for you!

So, what is the Quadratic Formula?

I will go over how to solve it, but first, the it looks like this:

quadratic formula

You can use this for any quadratic expression of the form ax2 + bx + c = 0, where “a” does not equal zero. (If you think about this condition, you can see that if a = 0, then there is no x2 term at all, and you are left with a linear equation or something of a higher order. Also, if a = 0, the quadratic formula then has 2(0) in the denominator, which equals 0 and causes the whole expression to be undefined. So, hopefully that short explanation will help you to remember that if a = 0, you cannot use the quadratic formula!)

Working through the math of the quadratic formula isn’t as difficult as you may think. To start, all you do is arrange your question into the form of ax2 + bx + c = 0, and then you can easily identify the coefficients for a, b, and c. Then, you simply substitute those values into the quadratic formula, and do the math. One thing to draw your attention to though is the “plus/minus” sign. Basically, the quadratic formula is really TWO formulas, one with a “-b + √…..” and one with “-b – √…..” These two formulas are what give you your two roots.

You will study this more in the future, but for now you may find it interesting that a quadratic equation, i.e. an equation with an x2 term, defines a parabola. The equation of all parabolas have x2 as the highest order exponent. As a result, you can imagine that a parabola drawn on an X-Y graph will cross the x-axis twice (at the most). These are the roots or solutions of the equation, and so that is why you cannot have more than 2 roots. Similarly, you can figure out why there may be 1 or even 0 roots, depending on where the parabola is located on the graph.

So, now that you know the answer to ‘what is the quadratic formula,’ next I will show you how to use it. Examples coming in my next post….