Quadratic Formula

Directions: What are the maximum and minimum values for c if x^2 + 12x + 32 = (x+a) (x+b) + c?

Hint

Hint

1) Can c be positive? Negative? Zero?

2) Try multiplying the binomials. Do you see any relationship betwee, a,b, and c?

3) Try graphing the function using this: https://www.desmos.com/calculator/4bpceyyd5j

Answer

Answer

One of the values can be found by splitting a and b equally as (x+6)(x+6) which gives c=-4. This is the minimum value for c.

There is no maximum value because the further apart a and b are in value, the higher the value of c to balance the equation. (Allowing a or b to be negative makes the upper limit infinite. For example a =1,000,000 b = -999,988.
As long as the sum is 12.)

Source: Jedidiah Butler

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3 comments

  1. Yes, the further apart a and b are in value, the higher the value of c. However the term 12x must be satisfied; this places limitations on c (sure c is a constant but it is still effected by 12x)
    11+ 1 = 12 (11) (1) = 11 32-11 = 21
    c has a maximum value of 21 if we are only considering whole numbers

    If we consider fractional numbers there is still a max for c
    –>11.9 + .1 = 12 (11.9) (.1) = 1.19 32- 1.19 = 30.81

    –>11.999 + .001 = 12 (11.999) (.001) = .011999 32 – .011999 = 31.988001

    as the value of the difference of a and b increases, the value of c gets closer to 32, but can never be greater than or equal to 32.

    The range of c is [-4, 32)

  2. Allowing a or b to be negative makes the upper limit infinite. For example a =1,000,000 b = -999,988.
    As long as the sum is 12.

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