How do you solve a quadratic equation by completing the square?

Answer 1

First I will describe the simpler case of the quadratic equation ax2+bx+c=0 where x=1.

To solve the quadratic equation by completing the square when a=1, first we subtract c from both sides to get x2 + bx = -c. We then rearrange the LHS of the equation into the form (x + b/2)2 however, this isn't quite right. If we expand out this form we get x2 + bx + b2 and we have an extra term of b2 from nowhere, therefore when we go into this new form we need to subtract b2 to 'complete the square'.

Now we have (x+b/2)2-b2=-c and it is just a simple case of rearranging and taking a square root to get the desired form of x= -b/2 ± sqrt(b2-c)

Now for the slightly more complicated case of where a is not 1.

With a not being 1 we cannot go from ax2 + bx to (sqrt(a)x + b/2) as when we expand it instead of getting bx we get sqrt(a)bx, so we have to use a different method. What we do is take a out of the expression as a factor to get a(x2 + bx/a + c/a) now we can do what we did before to get (x + b/2a)2 and this time we need to subtract b2/4a2. This gives us a((x + b/2a)2 + c/a - b2/4a2) multiplying a back in gives us a(x + b/2a)2 + c - b2/4a. Rearranging this for x ends up giving us the formula for solving quadratic equations, x = (-b ± sqrt(b2 - 4ac)) / 2a

I'll give an example of this latter case since all the letters can be a bit confusing when learning this technique.

example:

2x2 + 16x + 6 = 0

2(x2 + 8x + 3) = 0 taking out a factor 2

2((x + 4)2 - 13) = 0 completing the square

2(x + 4)2 - 26 = 0 factoring 2 back in

2(x + 4)2 = 26 rearranging

(x + 4)2 = 13 dividing by 2

x + 4 = ±sqrt(13) taking square root

x = -4 ±sqrt(13) rearranging for x

Verifying with the formula:

x = (-16 ±sqrt(162-48)) / 4

x = (-16 ±sqrt(208)) / 4

x = -4 ±(sqrt(208)/sqrt(16))

x = -4 ±sqrt(13)

The above shows why it is important to know the completing the square method for cases when you don't have a calculator handy, it is much easier to use the completing the square method than to work out what 162-48 is and many will not recognise that 208 is divisible by 16.