What is a quadratic equation and what do you have to do with one?

A quadratic equation is in the form of ax2 + bx + c = 0. Notice that the first term in this equation is x raised to a power of 2, or squared. Therefore, by definition, that is a quadratic equation. You may have to solve such an equation, or, in other words, determine values for x.

A review of the quadratic formula

One way to solve a quadratic equation is to use the quadratic formula. This is given as:

x = -b ± √(b^2 - 4ac) all over 2a


This may look unfriendly, but it is so useful that it is worth committing to memory.

Let’s take an example of a quadratic equation from the Official Guide: 

2x2 - x - 6 = 0

First identify the values of a (= 2), b (= -1), and c (= -6). Notice that if it is a subtraction, then the value is negative. Now substitute those values into the quadratic formula and simplify: 

Steps in solving. Simplifying gets to x = (1±7)/4
You may not be sure what '±' means, but it is actually fairly simple. It is just a quick way to write that there are two options, one using addition and the other using subtraction. Each of these cases will give a different solution.

If you use addition, you get x = 8/4 = 2

If you use subtraction, you get x = -6/4 = -1.5

We have therefore calculated two possible values for x and have solved the quadratic equation. The quadratic formula will always work for solving a quadratic equation, but in many cases there is an even faster method of solution called factorization. 


Take another quadratic equation as an example:

x2 + 7x + 12 = 0

Factorizing this equation means finding what two things, when multiplied together, will bring you to the equation. The first thing you can do is to open two sets of brackets. These will contain the two factors:

( ____ ) ( ____ ) = 0

The first term of our quadratic equation is x2. Place an “x” at the start of each bracket:

(x ___ ) (x ___ ) = 0

Now look at the sign in front of the third term (12). This is positive. We are looking for two numbers that multiplied together will bring you to 12. These two numbers must be either both positive or both negative. To determine which of these situations you have, look at the sign in front of the second term (7). Those numbers need to add to 7, which is also positive. Therefore, you know that the signs in our two brackets will both be positive: 

(x +  __ ) (x +  __ ) = 0

Now we are looking for two numbers that, multiplied together, will bring you to 12 and, added together, will bring you to 7. These two numbers are 4 and 3:

(x + 4) (x + 3) = 0

We now have the two factors of the equation. We have two factors that, when multiplied together, will bring you to 0. The only way that can happen is for one or both of the factors to equal 0. To make the first factor 0, x must be -4. To make the second factor 0, x must be -3. We have therefore solved the quadratic equation and found two possible values for x. These are -4 and -3.

The first time you do this it may seem difficult. However, once you practice it a little, it is definitely the fastest way to solve quadratic equations.