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GCSE Maths
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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.
Important point about square roots: 62 = 36. But also (-6)2 = 36 because -6 × -6 = + 36 (a minus × a minus = a plus). Therefore there are two square roots of 36: +6 and -6. We call 6 the positive square root of 36 and -6 is called the negative square root of 36.
So if x2 = 36, then x = +6 or -6 (since squaring either of these numbers will give 36).
However, the square root sign means "positive square root". So √36 = + 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 ± √( 52 - 4×3×(-8))
6
= -5 ± √(25 + 96)
6
= -5 ± √(121)
6
| = | -5 + 11 | or | -5 - 11 |
|
6 |
6 |
∴ x = 1 or -2.67
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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