(m) The difference between half a number and one-third of the same number is 69. What is the number?
(n) A man is twice as old as his son now. If he was three times as old as his son 10 years ago, how old is each now?
5.2 Quadratic equations
A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0, where a, b, c are fi xed real numbers and a ≠ 0. a is called the coeffi cient of x 2 and must not be zero. (Why?) b is called the coeffi cient of x. c is called the constant term.
If the highest power of the variable in an equation is 2, the equation is a quadratic equation.
Methods of solution There are two methods of solving quadratic equations:
1. Factorisation 2. The quadratic formula
1. Factorisation
Factorisation is the fastest way of solving quadratic equations. However, it does not always work. It depends on a fundamental idea in mathematics called the zero factor property, which states that if pq = 0, then p = 0 or q = 0, or both are equal to zero.
WORKED EXAMPLE
(3x – 2) p
(2x + 1) q
= 0
By the zero factor property: 3x – 2 = 0 or 2x + 1 = 0 3x = 2 3
x = 2 _
Solving a quadratic by factorisation
Solve 6x 2 – x – 2 = 0. This is a sum of three terms. Convert it into a product of two linear factors:
1. These are the solutions or roots of the equation 6 x 2 – x – 2 = 0. They are the only numbers in the universe for which 6 x 2 – x – 2 = 0.
2. They are the x co-ordinates of the points at which the curve y = 6 x 2 – x – 2 crosses the x-axis.
