Algebra – Improving Your Skills:

The Quadratic Formula and how to derive it

You can improve your algebra skills by learning to do a proof such as the derivation of the quadratic formula. For many of you this will be a challenging exercise but is well the effort. In the derivation below I have tried to include every step with a clear explanation for each.

In many algebra textbooks we may well see something like:
The question is: “How do we get from , which is the general quadratic equation to
, which is the quadratic formula?”

The answer is that in order to solve the general quadratic equation we have to employ a sequence of algebraic steps that will enable us to isolate x on one side of the equation.

So let’s start out with the general quadratic equation and define its various parts.

• x is called the independent variable which means that for any quadratic equation it can have 2, 1 or no real values depending upon the value of b -4ac , which is called the discriminant
• a ,b and c are called the coefficients and for each quadratic equation have constant real number values. The expression means that a ,b and c are members of the set of real numbers. The only restriction is that a is not allowed to be zero; if it were then the quadratic equation would not exist as we would be left withwhich is a linear equation.

To solve the general quadratic equation we need to follow the following algebraic steps:

1. Divide each term of the general quadratic equation by the coefficient a :
   and simplify each term so that we obtain:
2. Now we isolate the terms in x² and x on the left hand side of the equation by subtracting the
   term from both sides as follows:
   
   
3. The next step is the one that many students find to be difficult. We need to make the left hand side of the equation into a perfect square (like 1, 4, 9, 16 …). To do this we employ an algebraic trick in which we add the square of half the coefficient of x to both sides of the equation (note that the coefficient of x is now ):
   
   
4. Since the left hand side of the equation is now a perfect square it will factor as shown below and we can also simplify the right hand side of the equation:
   
   
   [ If you find the factoring process on the left hand side of the equation difficult to follow look at the following steps:
   and to convince yourself that the last of these steps works, consider:
   
5. Now continuing to simplify the right hand side of the equation we obtain:
   
6. The next step is to take the square root of both sides of the equation and remember to include a ± sign on the right hand side:
   
7. Further simplification of the right hand side produces:
   
8. Now we subtract from both sides of the equation:
   
9. Finally we add the two terms on the right hand side of the equation together:
   , which is the quadratic formula.

I hope that you were able to understand each of the steps in the above derivation. You should try to practice writing out the derivation from memory while always ensuring that you fully understand each and every step. In this way your algebraic skills will improve and you will soon be able to miss out some of the steps that start to become too obvious to you. It is vital that you continue to practice algebraic techniques. Experience as a mathematics teacher has long shown me that the students who have difficulty with calculus are those whose algebra skills are weak.

Time spent practicing your algebra now, particularly on challenging exercises like the one presented above will be time well spent!