Discriminants and determining the number of real roots of a quadratic equation

To understand the roots of quadratic equations, we must first ensure we understand the quadratic equation itself. This can be much simpler than other mathematical concepts, such as derivatives, which is as follows:

Ax^2 + bx + c = 0

Typically, these will equate to zero, but there are some instances where it won't. We don't need to worry about that now. Let's just focus on the roots, which are the values of x in the equation. These can relate to real or imaginary numbers. Once this is understood in our minds, we can start looking at what the discriminant formula looks like. Then, we will know what root to use, what equation to consider and what function to use. It will also tell us what is distinct and what will repeat. 

What is the discriminant formula? 

This can also be known as the quadratic equation roots formula. Let’s take a closer look at the formula itself.

∆ = b^2 – 4ac

Remember that the lettering here is in the quadratic equation. The b being the coefficient of the x term that is not squared. The a is the coefficient of the x term that is squared. The c is the same constant term in the equation without an accompanying x. So, make sure you understand where you need to look for these in the quadratic formula, or you will be lost with the discriminant formula when you try to work out the roots of quadratic equations.

This symbol ∆ in the quadratic equation roots formula stands for delta or the change. You can learn many of these symbols when you look closer at our maths course.

Now, it needs to be broken down into each case possibility to find the precise and correct answer exactly. 

∆>0: The function of this symbol is to say the discriminant is positive and has two distinct real roots. You can normally use the quadratic formula here to calculate the roots exactly. 

∆ = 0: In this case, the discriminant is zero. You will only find one real root, which will constantly repeat itself. When you try to use the quadratic formula here, it will simply give the value two times. 

∆ < 0: In our final case here, the discriminant is negative. That means the roots are non-real or imaginary. You're going to get the imaginary number here, even when you use the quadratic formula. You won't see the solution on a regular number line but only on a complex number line. 

Why is it important to understand this? 

When it comes to maths, at times, it may seem we're just memorising a suite of formulae to procure answers for an examination. In reality, when we start to dive deeper into this, such as looking closer at the discriminant, we will understand what the quadratic equation is all about. 

Before the equation is even solved, you can tell what the outcome is going to be and how many roots we'll see. It will also tell us what type of roots we're going to have as well. It's to make complex math much easier to digest and gives us a brief view and assessment. This is a guiding light in the complexity of quadratic equations. 

It must also be mastered for A levels if you want to gain high marks. It will allow you to work faster when you see similar equations, and at the same time, it will help you handle roots much faster. 

Maths is interconnected

Remember, this is just a small part of the maths you'll need to learn. Quadratic equations and relevant discriminant formulae are just a portion of the overall puzzle. You want to make sure that you understand both the core formula and the formula related to the delta itself. This will help you conquer algebra as it appears on the examination.