How do you get from one formula to another?

Question

Alright, well, I need to know how to get from
x^2+bx+c to x=-b (plus over minus sign +_) *square root* b^2-4ac/2a.
Can anyone help me? X_X

And, if anyone knows what this formula conversion is called, it would be of great help :)

Answer 1

Derivation of the quadratic formula by completing the square.

ax^2 + bx + c = 0
x^2 + (b/a)x + (c/a) = 0
x^2 + (b/a)x = -c/a
x^2 + (b/a)x + b^2/(4a^2) = b^2/(4a^2) - c/a
(x + b/2a)^2 = b^2/(4a^2) - 4ac/(4a^2)
x + b/2a = ±√[ b^2/(4a^2) - 4ac/(4a^2) ]

x = -b/2a ± √[ b^2/(4a^2) - 4ac/(4a^2) ]
= -b/2a ± √[ b^2- 4ac] / 2a

x = {-b ± √[ b^2- 4ac] } / 2a

Answer 2

The process is called completing the square, and it's easier when the coefficient of x^2 is 1. But that was just a slip in typing it out, wasn't it. You meant ax^2 + etc.

So, first step is divide each term by a so that x^2 has coefficient 1:

x^2 + (b/a)x + c/a = 0

Completing the square (when coefficient of x^2 is 1):
halve the coefficient of x, which gives us b/(2a), square it, giving (b^2)/(4a^2).
We want c/a out of the left side, so add -c/a to both sides. We want (b^2)/(4a^2) on the left side, so add it to both sides.

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

Now the left side is a perfect square, it's the square of
x + b/(2a)
and the right side, converted into a fraction with denominator 4a^2, becomes
(b^2 - 4ac)/(4a^2)

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

Take the square root of both sides, remembering to do +-:

x + b/(2a) = +or- sqrt((b^2 - 4ac)/(4a^2))
which = +or- sqrt((b^2 - 4ac))/(2a) [here we've taken the sq rt of the denominator and it's no longer under the square root sign]

Add -b/(2a) to both sides and you have the well-known formula!

And after typing all that, I look at the answer from novangeli..., which is the most sensible and deserves the 10 points!

Answer 3

The original formula you have above is
ax^2+bx+c=0, a quadratic.
To solve it for x, you apply the quadratic formula:

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

Answer 4

It is called the quadratic formula. Once you know actual number values for a, b, and c, you plug them into the formula to get the number value of x.

Answer 5

You want the derivation of the qudratic equation.

I'd type it here, but it wouldn't look as clear as in my source.