Do you remember the general form of the quadratic
equation?
It’s ‘AX squared
plus BX plus C’ equals zero!
And the coefficient of X squared which is
A, cannot
be zero!
ax 2 + bx + c = 0
What we’re looking for is a general formula
which gives us the value of x in
terms of A, B and C. Let’s use the method
of completing the square to solve for
the value if X.
To make the first term a perfect square, let’s
divide the entire equation by A.
This is what we get:
x 2 + (b/a)x + c/a = 0
Looking at the variable part, we need to think
of a third term to add here,
which will again give us a perfect square.
‘2 times x times WHAT’ will give us ‘b
over a’ times x?
It will be ‘b over 2a’.
So
we add and subtract ‘b over 2a’ the whole
squared here.
x 2 + (b/a)x + (b/2a) 2 – (b/2a) 2 + c/a
= 0
If you did not understand what we did here,
make sure you have a look at our
‘Completing the Square’ tutorials.
It’ll give you a much better idea!
The first three terms together form a perfect
square.
Together it can be
written as ‘x plus, b over 2a’ the whole
squared.
(x + b/2a) 2 – (b/2a) 2 + c/a = 0
Now can we write these two terms as one term?
Yes, we can!
First, we write this as ‘b squared’ over
‘4 a squared’.
(x + b/2a) 2 – b 2 /4a 2 + c/a = 0
The LCM of 4A squared and A is 4 A squared.
So these two terms can be
written like this:
(x + b/2a) 2 – (b 2 – 4ac)/4a 2 = 0
Transposing this to the right hand side, will
give us this:
(x + b/2a) 2 = (b 2 – 4ac)/4a 2
If the right hand side is positive, we can
take the square root on both sides to
get this!
x + =
We’ve almost got the value of ‘x’ in
terms of a and b.
We just need to
transpose this term to the right!
x = -
Simplifying the right hand side will give
us this:
x =
Guess what, we have the roots of the quadratic
equation ‘AX squared plus BX
plus C’ equals zero!
They are negative B plus root of B squared
minus 4AC, over 2A…
AND
negative B minus root of B squared minus 4AC,
over 2A.
x = ,
And this can also be written like this:
x =
This formula for finding the roots of a quadratic
equation is known as the
quadratic formula!
In the next lesson, we will solve a couple
of quadratic equations using this
formula!
