1. ## Quadratics ( i think )

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 ?

2. 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: .$\displaystyle (11)(\text{-}11) \;=\;\boxed{\text{-}121}$

thank you