Quadratic equations

Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

Sketch the graphs for this implicitly defined family of functions.

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio!

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel lines are 1 unit and 2 units.

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.