You use the quadratic formula when youíve
got a quadratic equation to solve that you
canít factorise.
And to use the quadratic formula you need
to be dead familiar with the standard format
for quadratic equations ñ here it is.
You start with an x-squared term... then an
x term... then a number term... and itís
always equal to zero.
And it needs to be in this form, cos otherwise
your a, b and c wonít make sense.
And here is the lovely formula: x equals minus
b... plus or minus the square root of... b
squared minus 4ac, all over 2a.
Beautiful.
And hereís an example.
Now, itís a good idea to jot down the values
of a, b and c, but before we do that letís
just check that itís in the standard format.
So weíve got an x-squared term, then an x
term, number term, equal to zero.
So thatís all fine.
So whatís ëaí equal?
Well ëaí is the number before x-squared,
but we havenít got one, so it must be 1 x-squared,
so a = 1.
b = 2.
And c = -3.
OK off we go.
x equals minus b (-2), plus or minus the square
root of... b squared (thatís 2 squared),
minus 4 a c.
OK, so itís 4 times ëaí, which is 1, times
ëcí which is -3.
And itís all over 2a, which is 2 times 1.
Lots of people seem to put an ëaí rather
than ë2aí, so donít forget ñ itís all
over ë2aí.
Donít be one of those silly people.
OK, now what does this square root sign give
us?
Thereís two squared, which is 4... and then
inside this bracket weíve got 4 times 1 which
is 4, times -3, so thatís -12.
So weíve got minus... minus 12, so we need
to be really careful with those minus signs...
really, really careful actually.
Actually really, really, really careful...
and so on.
And thatís all over 2 times 1, which is 2.
So now in this square root sign weíve got
4 minus minus 12, so thatís 4 plus 12 which
is 16, and the square root of 16 is 4.
Now weíre going to have two solutions to
this: weíve going to have -2 plus 4 over
2, and -2 minus 4 over 2.
So if itís plus 4 weíve got -2 plus 4 over
2, which is 2 over 2, and that equals 1.
Or weíre going to have -2 minus 4, so thatís
-6 over 2 and that equals -3.
So theyíre our two solutions, 1 or -3.
OK another example, solve this... ëGive your
solutions to two decimal placesî.
Now when youíre in the exam you might get
a quadratic equation to solve and you need
to decide whether to factorise it, or complete
the square, or use the formula.
Well a good rule of thumb is to use the formula
when youíre told the solutions are not whole
numbers... and here youíre told to give your
solutions to two decimal places, so they canít
be whole numbers.
So the formulaís your best bet.
So letís make sure itís in the standard
format...
x-squared term, x-term, number term, equal
to 0.
Now, a = 3, b = -2, and c = - 7.
So weíve got two minus signs here, so weíre
on high alert... alright, now into the formula:
x equals minus b... so weíve got minus minus
2, which is plus 2... then plus or minus the
square root of... b squared, so thatís -2
squared... minus 4ac, so thatís 4 times ëaí
which is 3, times ëcí which is -7... all
over 2a, so thatís 2 times 3.
OK, whatís in the square root sign?
Weíve got -2 squared ñ well, -2 times -2
is 4.
And then in the bracket weíve got 4 times
3 (which is 12), times -7... well 12 times
7 is 84, times -7 is -84.
So weíve got minus minus 84, and thatís
over 6.
So now youíve got 4 minus minus 84, so thatís
4 plus 84, 4 + 84 is 88, so itís root 88.
And now you canít do anything with this,
so itís time to get your calculator out.
Now you do 2 plus root 88 over 6, and that
gives you that when rounded to two decimal
places.
And then you do 2 minus root 88 over 6, and
that gives you that when you round to two
decimal places.
So they are your two solutions.
So thatís it on the quadratic formula ñ
itís basically just a question of plugging
in the numbers.
But you do need loads of practice so youíre
really comfortable actually using it, and
remember to take extra special care with minus
signs.
