Quadratic equations

problem
Golden ratio

Age

16 to 18

Challenge level

Solve an equation involving the Golden Ratio phi where the unknown occurs as a power of phi.
problem
Power quady

Age

16 to 18

Challenge level

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
problem
Pent

Age

14 to 18

Challenge level

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.
problem
Pareq calc

Age

14 to 16

Challenge level

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.
problem
How old am I?

Age

14 to 16

Challenge level

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?
problem
Always two

Age

14 to 18

Challenge level

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.
problem
Cocked hat

Age

16 to 18

Challenge level

Sketch the graphs for this implicitly defined family of functions.
problem
Darts and kites

Age

14 to 16

Challenge level

Explore the geometry of these dart and kite shapes!
problem
Target six

Age

16 to 18

Challenge level

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

Age

14 to 16

Challenge level

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