Open Middle®
Directions: What are the maximum and minimum values for c if x^2 + 12x + 32 = (x+a) (x+b) + c?
Hint
1) Can c be positive? Negative? Zero?
2) Try multiplying the binomials. Do you see any relationship between, a,b, and c?
3) Try graphing the function using this: https://www.desmos.com/calculator/4bpceyyd5j
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
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)
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.
Thank you for providing examples to clarify the explanation for the upper limit.