If the highest power of the pronumeral in an equation is two, then it is said to be a quadratic equation.
Solving Quadratic Equations
We will consider the following methods to solve a quadratic equation:
- Factorising and using the Null Factor Law
- Quadratic formula
The Null Factor Law
From this we can infer that:
If the product of any two numbers is zero, then one or both of the numbers is zero.
That is, if ab = 0, then a = 0 or b = 0 (which includes the possibility that a = b = 0).
This is called the Null Factor Law; and we use it often to solve quadratic equations.
Example 1
Solution:
Equations Involving Factorisation
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To solve a quadratic equation, make the right-hand side of the equation zero by transposition. Then factorise the left-hand side, if possible; and then use theNull Factor Law.
Example 2
Solution:
Note:
Example 3
Solution:
Example 4
Solution:
Sometimes the solution of an equation can be simplified by the use of substitution, as illustrated in the following example.
Example 5
Solution:
Quadratic Equations Containing Fractions
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The solution of an equation containing fractions is obtained by multiplying all terms by the lowest common denominator, making the RHS equal to zero, factorising the LHS and using the Null Factor Law.
Example 6
Solution:
We write the solution as follows:
Example 7
Solution:
Use of the Difference of Two Squares
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Recall that:
A solution of a quadratic equation may be obtained by using DOTS, as illustrated below.
Example 8
Solution:
Recall that:
A quadratic equation can be obtained by using the square completion method, as illustrated below.
Consider the quadratic equation:
 This is called the quadratic formula.
Example 9
Solution:
Alternative way:
Note:
- If the LHS of an equation can not be factorised, then we use the quadratic formula.
- The quadratic formula can be used to solve any quadratic equation.
Example 10
Solution:
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