| This takes the quadratic equation and breaks it into manageable factors.
There are several ways to factorise Q equations
Completing the Square is one | Completing the Square is very easy if you remember that ...
The bit in the brackets is ALWAYS equal to
a(x + b/2a)
So if you get something really nasty looking like
4x2 + 3.2x + 10.4
you just say a = 4 and b = 3.2
so the answer to the squared bit is
4(x + (3.2)/8) = 4(x + 0.4)2
We now need to find the added bit on the end
by calling it d and making the new squared equation = the starting equation, we get ...
4(x + 0.4)2 + d = 4x2 + 3.2x + 10.4
making
4x2 + 3.2x + 6.4 + d = 4x2 + 3.2x + 10.4
so
d = 10.4 - 6.4 = 4
giving us a completed square of
(2x + 0.8)2 + 4
and if you want to know what x is, well ...
(2x + 0.8) = Square root of 4
(remember that the square root of a number can be either positive or negative)
so (2x + 0.8) = -2 or (2x + 0.8) = +2
so x = -1.4 or 0.6 | Minimum value
Make the part in the brackets equal to zero
When the squared part is equal to zero, f(x) reaches its minimum value
f(x) = (x+4)2-14
f(-4) = (-4+4)2-14
f(-4) = (0)2-14 = -14
so the minimum value here is -14 |
| notes
A "normal" is another name for a perpendicular line from a tangent.
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