What is the minimum product of two numbers whose difference is 22? What numbers yield this product?
FYI i asked my teacher and she didn't quite explain it the way i hoped so i was hoping someone could explain it better ?
Hello, diehardmath4!
What is the minimum product of two numbers whose difference is 22?
What numbers yield this product?
Let $\displaystyle x$ = the larger number.
Let $\displaystyle y$ = the smaller number.
Their difference is 22: .$\displaystyle x - y \:=\:22 \quad\Rightarrow\quad y \:=\:x-22$ .[1]
Their product is: .$\displaystyle P \:=\:xy$ .[2]
Substitute [1] into [2]: .$\displaystyle P \;=\;x(x-22)\quad\Rightarrow\quad P \:=\:x^2 - 22x$
This is an up-opening parabola; its minimum is at its vertex.
The vertex is at: .$\displaystyle x \;=\;\frac{\text{-}b}{2a} \;=\;\frac{22}{2(1)} \quad\Rightarrow\quad \boxed{x\:=\:11}$
Substitute into [1]: .$\displaystyle y \:=\:11-22\quad\Rightarrow\quad\boxed{ y \:=\:\text{-}11} $
The minimum product is: .$\displaystyle (11)(\text{-}11) \;=\;\boxed{\text{-}121}$