Quadratic Equations - Revision Notes

In these revision notes for Quadratic Equations, we cover the following key points:

• What are quadratic equations?
• What are coefficients and the constant in a quadratic equation?
• How do we solve quadratic equations by factorization?
• How many methods of factorization are used to solve quadratic equations?
• What kind of quadratic equations are best to solve by factorization?
• How can we solve a quadratic equation by completing the square?
• What are the advantages and disadvantages of each of these methods?

Quadratic Equations Revision Notes

A quadratic equation is a second - order equation with one variable that has a general form

It is called a 'second - order equation' because its variable x is in the second power. On the other hand 'with one variable' means that only the variable x is unknown; the other letters represent known numbers, where a and b are called coefficients while c is a constant.

Solving a quadratic equation means finding the value(s) of the variable x (we call them 'roots') for which the equation becomes true.

There are several methods for solving a quadratic equation. Some of them are:

1. Factorizing the left side of the original equation in the form
   (px - m)(qx - n) = 0
   where
   p ∙ q = a
   m ∙ n = c
   and
   - np - mq = b
   are relations involving new coefficients m, n, p and q to be found mostly by guessing.
2. When the quadratic equation has no constant (c = 0), it is better to factorize it as
   ax² + bx = x(ax + b) = 0
3. Factorizing the left part of the equation so that the part containing the variable takes the form of one of the first two special algebraic identities. The rest must also fit this form but in the first power. In other words, we must try to express the equation in the form
   (kx ± t)² ± r(kx ± t) = 0
   where k = √a while t and r are numbers. After doing this, we may make a new factorization in the form
   (kx ± t) ∙ [(kx ± t) ± r] = 0
   Completing the square. We must therefore try to express a given quadratic equation
   ax² + bx + c = 0
   in the form
   (x + p)² + q = 0
   where p and q are numbers.

Completing the square gives us the possibility to solve the quadratic equation by solving two first - order equations with one variable.

The procedure applied to complete the square is as follows:

Step 1: First of all, we find the expression in the brackets. For this, we must express the squared part in the form

where b is the coefficient preceding x in the original equation. Thus, it is obvious that p = B/2.

Step 2: Expanding the last expression yields

Step 3: Check the difference between the old and new constant.