Expert: Paul Klarreich - 10/8/2006

Question
My name is Pamela and I am studying Per-Cal.  I really don't even know how to start is question, so I would love any help you can give.

By completing the square on ax2 + bx + c = 0, derive the quadratic formula. That is, starting with the equation 0 = ax2+bx+c, show that the zeroes of this function are found by the formula x = −b ?} ?ãb2 − 4ac
            2a  

Answer
pamela Asks in Category Calculus ...
 
Subject:  quadratic function
Private:  no
 
Question:  My name is Pamela and I am studying Per-Cal.  I really don't even know how to start this question, so I would love any help you can give.

By completing the square on ax2 + bx + c = 0, derive the quadratic formula. That is, starting with the equation 0 = ax2+bx+c, show that the zeroes of this function are found by the formula [YES, I KNOW]
................................................
Hi, Pamela,

Somehow your character set does not come through well on the computer, but I know what you are asking.  The derivation of the QF can be found in practically any intermediate algebra text, and goes like this:

WARNING: THE FOLLOWING DISCUSSION MAY CONTAIN FRACTIONS AND OTHER MATERIAL INAPPROPRIATE FOR CERTAIN COMPUTING SYSTEMS.  BE SURE TO VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.

ax^2 + bx + c = 0   << Now divide by a, to get an equation that starts with x^2

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

x^2 + (b/a)x         = - c/a

The coefficient of x is b/a.  To complete the square, always take half the coefficient of x and square it, adding that to both sides.

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

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

It's time to write as better-looking fractions:
      b        b         b^2     c
x^2 + --- x + (---)^2  = ----- - ---
      a       2a        4a^2     a

Now write the LHS as a square, and combine fractions on the right side; the LCD is 4a^2.

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

Beginning to look familiar?  Now take square roots, and here is where that +- comes in.
    b         sqrt(b^2 - 4ac)
x + --- =  +-  ---------------
   2a              2a

Solve by subtracting  b/2a on both sides:

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

Now the right side has a common denominator of  2a, so

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

That's your formula.  (Two formulas, actually, when there is a +- in it.)