NOTE: If x² = 36, then x = +6 or -6 (since squaring either of these numbers will give 36).
Completing the Square
9 and 25 can be written as 3² and 5² whereas 7 and 11 cannot be written as the square of another exact number. 9 and 25 are called perfect squares. Another example is (9/4) = (3/2)². In a similar way, x² + 2x + 1 = (x + 1)².
To make x² + 6x into a perfect square, we add (6²/4) = 9. x² + 6x + 9 = (x + 3)² . To complete the square in this way, we take the number before the x, square it, and divide it by 4. This technique can be used to solve quadratic equations.
Example:
Solve x² - 6x + 2 = 0 by completing the square
x² - 6x = -2
(to complete the square on the LHS (left hand side), we must add 6²/4 = 9. We must, of course, do this to the RHS aswell).
x² - 6x + 9 = 7
(x - 3)² = 7
(Now take the square root of each side)
x - 3 = ±2.646
so x = 5.646 or 0.354
Some people don't like this method and an alternative is to use the quadratic formula. This is actually derived by completing the square.
The Quadratic Formula:
Where the equation is ax² + bx + c
Example:
Solve 3x² + 5x - 8 = 0
x = -5 ±
( 5² - 4 x 3 x -8)
6
= -5 ±
(25 + 96)
6
= -5 ±
(121)
6
= -5 + 11 or -5 - 11
6 6
so x = 1 or -2.33
Factorising
Sometimes, quadratic equations can be solved by factorising. In this case, factorising is probably the easiest way to solve the equation.
There is no simple method of factorising an expression. One way, however, is as follows:
Example:
Factorise 12y² - 20y + 3
12y² - 18y - 2y + 3 (here the 20y has been split up into two numbers whose multiple is 36. 36 was chosen because this is the product of 12 and 3, the other two numbers).
The first two terms, 12y² and -18y both divide by 6y, so 'take out' this factor of 6y.
6y(2y - 3) - 2y + 3 [we can do this because 6y(2y - 3) is the same as 12y² - 18y]
Now, make the last two expressions look like the expression in the bracket:
6y(2y - 3) -1(2y - 3)
The answer is (2y - 3)(6y - 1)
Example:
Factorise x² + 2x - 8
We need to split the 2x into two numbers which multiply to give -8. This has to be 4 and -2.
x² + 4x - 2x - 8
x(x + 4) - 2x - 8
x(x + 4)- 2(x + 4)
(x + 4)(x - 2)
Once you work out what is going on, this method makes factorising any expression easy. It is worth studying these examples further if you do not understand what is happening. Unfortunately, the only other method of factorising is by trial and error.
Once you have factorised an equation, it is easy to solve.
Example:
Solve x² + 2x - 8 = 0
(x - 2)(x + 4) = 0
therefore, either x - 2 = 0 or x + 4 = 0
so, x = 2 or x = - 4
If you do not understand the third line, remember that for (x - 2)(x + 4) to equal zero, then one of the two brackets must be zero.
