quadratic equations ?

Question

There are 3 types of ways to solve it, right ? The 3 ways are factoring, quadratic formula, and completing the square ? How do you do the completing the square ? can you please explain and give an example ? please and thank you.

Answer 1

To complete the square, you move enough of the constant over to the side so that the constant term (c) that remains on the left-hand side makes a perfect square.

If you have ax^2 + bx + c as a perfect square, then it would be of the form:

{sqrt(a)x + (b/[2sqrt(a)])}^2

... which when multplied out equals

(sqrt(a)*sqrt(a)x^2) + (2 * sqrt(a)b/[2sqrt(a)])x + (b/[2sqrt(a)])^2
ax^2 + bx + (b^2/4a)

... which means that c must equal (b^2/4a) to have a perfect square. So, whatever c is in your equation, you add or subtract to leave (b^2/4a) on the left.

Let's say you are completing the square to solve:

x^2 + 6x + 1 = 0

You want 6^2/4*1 = 36/4 = 9 on the left, so you add eight to both sides:

x^2 + 6a + 1 = 0
x^2 + 6x + 9 = 8
(x+3)^2 = 8

Now that you have a perfect square, you take the square root to solve:

sqrt( (x^3)^2 ) = sqrt(8)

x+3 = +/- sqrt(8) = +/- 2sqrt(2)

And this gives us the two solutions:

x + 3 = 2sqrt(2) --> x = -3 + 2sqrt(2)
x + 3 = -2sqrt(2) --> x = -3 - 2sqrt(2)

If you had used the quadratic to solve the same equation, (a=1, b=6, c=1), you'd get:

x = ( -b +/- sqrt(b^2 - 4ac) ) / 2a
x = ( -6 +/- sqrt(36 - 4*1*1) ) / 2*1
x = -3 +/- sqrt(32)/2
x = -3 +/- 4 sqrt(2) /2
x = -3 +/- 2 sqrt(2)

... the same answer.

Answer 2

say if ur equation isnt factorable like uhmm

x^2-4x+6 =0
completeing the square would be dividing the middle term by a half and squaring giving u (4/2)^2=4

so ur last term has to be 4 but u have 6

sooo haha you subtract two from the 6

giving u (x^2-4x+4)-2=0
(x-2)^2=2

hope that helped

Answer 3

Suppose you have ax^2 + bx + c =0

You try to find a square that involves ax^2+bx
that would be:
(sqrt(a)x + g) ^2 = x^2 + 2sqrt(a)gx + g^2
you should have 2sqrt(a)g = b so g= b/2sqrt(a)
so g^2 = b^2/4a
So you write the whole thing as:
(ax+g)^2 - g^2+c =0
And now you find ax+g = +-sqrt(g^2-c)
And you carry on to find the x.