A. The Golden Ratio
I have a line divided into two unequal sections so that the ratio of the shorter length to the longer length is the same as the ratio of the longer length to the whole length of the line. What is the ratio?
Solution
Call the lengths of the two sections A and B. A/B=B/(A+B). Cross-multiply: A^2+AB=B^2, so A^2+AB-B^2=0 and, using the quadratic formula, A=(-B+sqrt(B^2+4B^2)/2=(-B+Bsqrt(5))/2=B(-1+sqrt(5))/2. Therefore the fraction A/B=(sqrt(5)-1)/2=0.618 or -1.618 approx. The positive root applies because A and B are considered positive lengths.
An alternative solution is to let r=A/B, which is the ratio, then r=1/(r+1); r^2+r-1=0 and r=(sqrt(5)-1)/2.
This Golden Ratio keeps popping up in different contexts and seems to be aesthetically pleasing.
B. The A sizes
A rectangular piece of paper is folded in half so that the ratio of length to width remains the same after folding as before folding. What is the ratio?
Solution
If A and B are the length and width, the area is AB, and after folding in half the area becomes 1/2(AB). In the process of folding, only one side (choose the length A) is halved so the new dimensions of the rectangle are A/2 and B, but the ratio of the sides is still A/B, and the length and width are B and A/2 (the length has become the width and the width has become the length). So A/B=length/width=B/(A/2)=2B/A. If A/B=2B/A then A^2=2B^2. The ratio of the sides is A/B which we'll call r. So r^2=2 and r=sqrt(2)=1.414 approx. So the length is 1.414 times the width. The paper we use today is largely based on this ratio. A4 is the standard size used for printers. A3 is larger and tends to be used for drawings and posters. A5 is smaller and tends to be used for booklets and leaflets. All use the square root of 2 as the ratio of the sides.
C. Identity
Prove that the difference between the cubes of two consecutive integers is one more than three times their product.
Solution
The difference between the cubes of consecutive integers (x+1)^3-x^3 where x is an integer. The product of consecutive integers is x(x+1). The difference between the cubes is x^3+3x^2+3x+1-x^3=3x^2+3x+1=3x(x+1)+1 which is three times the product of the integers plus 1.
These questions use in different ways a quadratic expression. Although this reply is later than you wished for, what goes round comes round, and you may have the opportunity of using the ideas.