Quadratic Equations

A quadratic equation is an equation where the highest power of x is x². There are various methods of solving quadratic equations, as shown below.

NOTE: If x² = 36, then x = +6 or -6 (since squaring either of these numbers will give 36). However, Ö36 = + 6 only.

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. The resulting expression, x² + 6x + 9 = (x + 3)² and so is a perfect square. This is known as completing the square. 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, as demonstrated in the following example.

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 also].
\ x² - 6x + 9 = 7
\ (x - 3)² = 7
[Now take the square root of each side]
\ x - 3 = ±2.646     (the square root of 7 is +2.646 or -2.646)
\ x = 5.646 or 0.354

Completing the square can also be used to find the maximum or minimum point on a graph.

Example

Find the minimum of the graph y = 3x² - 6x - 3 .

In this case, the x² has a '3' in front of it, so we start by taking the three out: y = 3(x² - 2x -1) .  [This is the same since multiplying it out gives 3x² - 6x - 3]
Now complete the square for the bit in the bracket:
\ y = 3[(x - 1)² - 2]
Multiply out the big bracket:
\ y = 3(x - 1)² - 6

We are trying to find the minimum value that this graph can be. (x - 1)² must be zero or positive, since squaring a number always gives a positive answer. So the minimum value will occur when (x - 1)² = 0, which is when x = 1. When x = 1, y = -6 . So the minimum point is at (1, -6).

Some people don't like the method of completing the square to solve equations 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 = 0

Example

Solve 3x² + 5x - 8 = 0

x = -5 ± Ö( 5² - 4×3×(-8))
                   6

  = -5 ± Ö(25 + 96)
               6

  = -5 ± Ö(121)
            6

\ 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.

Example

Solve x² + 2x - 8 = 0
\ (x - 2)(x + 4) = 0
\ either x - 2 = 0 or x + 4 = 0
\ 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.

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