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….

How to Solve Quadratic Equations – Part 2

This post continues from where my last post left off, on how to solve quadratic equations. I explained the general form that a quadratic equation will take, with the key being that there is an x2 term present. To solve them without using the quadratic formula, you need to use a bit of factoring methods to come up with the roots. In particular, one common factoring method to use is the grouping method of factoring. Then, once factored, you consider the property that says “two terms multiplied will equal zero only if one or both of those terms is 0.” This may seem like a lot of work, and may sound a bit confusing with all the steps you need to take. But I think with a bit of practice you will come to better appreciate and understand the process you need to follow to arrive at your solution. You will see that you already know the individual steps you need to solve the equation. You just need to become familiar with the order that you use these steps.

Follow along through my example and you will hopefully be able to see what I mean.

Let’s consider the equation x2 + 7x + 10 = 0

First, we can identify that there is an x2 term (with a non-zero coefficient… 1), so we can say that it is a quadratic equation.

To solve a quadratic equation, we want to determine the roots, or what values make the equation true. To help us to achieve this, we want to rearrange the left side so that it is a product of two terms (or expressions). In this way, we can say that “something times something equals zero”. And since we need one of those “somethings” to be zero if the product is zero, we essentially break this down to “something #1 = 0” and “something #2 = 0”, and by solving these two simpler equations, we will arrive at our roots. So, continuing with our example then, let’s factor it. Review my post on methods of factoring if you need a bit of a refresher!

x2 + 7x + 10 = 0
(x + 2)(x + 5) = 0

This is what we’re looking for: two expressions multiplied together to give zero. Now, we have two equations to work with to find our roots of the quadratic equation. Rewriting, this gives us:

x + 2 = 0 and x + 5 = 0

And quite obviously, these can be solved to show that x = (–2) and (–5). And since we followed that whole process, we can consider these two values to be roots of our original quadratic equation. However, it is VERY IMPORTANT to substitute these values back into the original equation to check! With these values, we can show that:

(–2)2 + 7(–2) + 10 = 0
4 – 14 + 10 = 0…….. this is true. So –2 is for sure one of the roots. I’ll leave –5 for you to verify on your own.

If you find a question and proceed all the way through to find the roots, and you go and plug them back into the original equation, if one of the roots does NOT satisfy the equation, you cannot count it as one of the roots. This sometimes happens when you have an expression in a denominator (eg. (x – 2)), and if you determine through the above steps that your expression gives you a root of 2, by plugging this into your original equation, specifically into the denominator, the denominator will equal 0 and cause the expression to be undefined. Therefore, this root does not satisfy the original equation and you just ignore it.

I hope this has helped to explain the process you need to follow to solve quadratic equations. With practice, they will become second nature. However, despite all of the work required, sometimes it just is not practical or apparent how to factor your quadratic equation. In these cases, you would likely want to rely on the use of the quadratic formula, which I will go over in a future post to explain what it is and how it works. Let me know if this makes sense or if you’d like anything more added.

How to Solve Quadratic Equations

Following up my previous post that gave you advice on how to solve equations, in this post I would like to go over some strategies on how to solve quadratic equations. Quadratic equations become very common in high school math and college math, and they require a bit more work sometimes to solve. You may already have experience using the quadratic formula, which I will explain shortly and is extraordinarily good to memorize! First though, let’s go over solving quadratic equations. To do this, you will commonly rely on factoring quadratics techniques. You can refer to my previous post on methods of factoring for some additional tips!

When most students hear “quadratic equation,” they usually get anxious because quite often this means having to work with the quadratic formula. This formula is more complicated than most that you have probably encountered up to this point, but factoring quadratics doesn’t always rely on the quadratic formula! In fact, they can be quite simple! A quadratic equation isn’t just “something that needs the quadratic formula” to solve it. Quite simply, a quadratic equation is just an equation that can written in this form:

ax2 + bx + c = 0 where a, b, and c are real numbers and a does not equal 0

See? That doesn’t sound so bad. 😉 The KEY is that the “a” value is not 0. b and c can be, but not a. You need to have the x2 term.

Remember, “solving an equation” means to find the roots or solution… or, what makes the expression true? To do this with quadratic equations, we rely on the property that says “two terms multiplied = 0 only if one or both of those terms is 0.” Remember this property! It is key to the quadratic factoring method. If we combine this property with our ‘grouping’ factoring method, you will see how this all comes together.

More to come in my next post, How To Solve Quadratic Equations – Part 2….