Completeing the Square?

Question

Please help! I am in DIRE need of instructions on how to "complete the square". Can anyone give me an example with step by step instructions on how to do it. I would really appreciate it, if someone took the time to explain it like I am doing it for the first time.
Thanks

Answer 1

Suppose you have 3x^2 +6x - 6 = 0
Divide by 3 to get x^2 alone getting:
x^2 + 2x -2 = 0
Now look at the x term. It is being multiplied by 2. You need to divide this by 2 getting 2/2 = 1 and then square the reult getting 1^2 =1. This is the value that will complete the square, and you need to add it to both sides of the equation getting:
x^2 +2x + 1 -2 = 1
Now x^2+2x +1 is a perfect square and so our equation is now:
(x+1)^2 +1-2=1
(x+1)^2 = 2
Now take sqrt of both sides getting:
x+1 = +/-sqrt(2)
and finally, x = -1 +/- sqrt(2)

If you had the general quadratic equation ax^2 +bx + c = 0
we would divide through by a getting:
x^2 +b/a x +c/a = 0
x^2 +b/a x = -c/a
Now take 1/2 of b/a getting b/2a, then square it getting b^/4a^2 and add it to both sides of the equation getting:
x^2 +b/a x +b^2/4a^2 = -c/a + b^2/4a^2
(x+b/2a)^2 = (-4ac + b^2)/4a^2
x+b/2a =+/- sqrt ((b^2-4ac)/4a^2)
x+b/2a =+/-sqrt(b^2-4ac)/2a
x = -b/2a +/- sqrt(b^2-4ac)/2a
x = [-b +/- sqrt(b^2-4ac)]/2a which is the quadratic formula derived by completing the square.

Hope this helps you. It will if you study it until you know it well.

Answer 2

First, the coefficent of x^2 must be 1. Divide by any number infront of x^2 1st

2x^2 + 8x - 6 = 0

x^2 + 4x -3 = 0.................x^2 + 4x = 3
Then take 1/2 the coefficent of x, and square it. Then add to both sides.

1/2(4) = 2...2 squared = 4

x^2 + 4x + 4 = 3 + 4

Factor the left side

(x + 2)^2 = 7
Take the sq rt of both sides..

x + 2 = +- sq rt 7

x = -2 +- sq rt 7

Answer 3

ok, lets say example of
x^2-6x+1=0
First bring the number to the other side
x^2-6x=-1
Now divide -6 by 2 = -3, then square -3 to get 9
(x-3)^2 = -1 + 9
square root both sides
x-3=+-sqrt(8)
x=+-sqrt(8) + 3

Answer 4

ax^2 + bx + c = 0

subtract "c" from both sides

ax^2 + bx = c

divide everything by "a"

x^2 + (b/a)x = (-c/a)

find half of (b/a), square it, add to both sides

x^2 + (b/a)x + ((b^2)/(4a^2)) = (-c/a) + ((b^2)/(4a^2))

factor left side into perfect square

(x + (b/(2a)))^2 = (-4ac + b^2)/(4a^2)

sqrt both sides

x + (b/(2a)) = ±sqrt((b^2 - 4ac)/(4a^2))

x + (b/(2a)) = (±sqrt(b^2 - 4ac))/(2a)

subtract (b/(2a)) from both sides

x = (-b/(2a)) ± (sqrt(b^2 - 4ac)/(2a))

x = (-b ± sqrt(b^2 - 4ac))/(2a)

this is also what is known as the quadratic formula.

Answer 5

there are a variety of proofs that it is not possible by construction to complete the square .. so I would move on to the next question.

Answer 6

What square. Please supply a bit more details, then i will try to do that for you.