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About Paul Klarreich
Expertise
All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience
I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

You are here: Experts > Teens > Homework/Study Tips > Calculus > quadratic function

Calculus - quadratic function

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.)

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