Search by Topic

Resources tagged with Quadratic equations similar to Infinite Continued Fractions:

Filter by: Content type:
Stage:
Challenge level: Challenge Level:1 Challenge Level:2 Challenge Level:3

There are 36 results

problem Icon

Continued Fractions II

Stage: 5

In this article we show that every whole number can be written as a continued fraction of the form k/[1+k/[1+k/etc.]].

problem Icon

Good Approximations

Stage: 5 Challenge Level: Challenge Level:1

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

problem Icon

Golden Thoughts

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

problem Icon

Golden Mathematics

Stage: 5

A voyage of discovery through a sequence of challenges exploring properties of the Golden Ratio and Fibonacci numbers.

problem Icon

Implicitly

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Can you find the maximum value of the curve defined by this expression?

problem Icon

In Between

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Can you find the solution to this algebraic inequality?

problem Icon

Conjugate Tracker

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Make a conjecture about the curved track taken by the complex roots of a quadratic equation and use complex conjugates to prove your conjecture.

problem Icon

Resistance

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Find the equation from which to calculate the resistance of an infinite network of resistances.

problem Icon

Plus or Minus

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem Icon

Golden Fibs

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem Icon

Darts and Kites

Stage: 4 Challenge Level: Challenge Level:1

A rhombus PQRS has an angle of 72 degrees. OQ = OR = OS = 1 unit. Find all the angles, show that POR is a straight line and that the side of the rhombus is equal to the Golden Ratio.

problem Icon

Pentakite

Stage: 4 and 5 Challenge Level: Challenge Level:1

ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE.

problem Icon

Golden Ratio

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem Icon

Golden Eggs

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem Icon

Polar Flower

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

This polar equation is a quadratic. Plot the graph given by each factor to draw the flower.

problem Icon

Root Tracker

Stage: 5 Challenge Level: Challenge Level:1

Track the roots of quadratic equations as you move the corresponding graphs and discover the transitions from real to complex roots.

problem Icon

What Are Complex Numbers?

Stage: 5

This article introduces complex numbers, brings together into one bigger 'picture' some closely related elementary ideas like vectors and the exponential and trigonometric functions and. . . .

problem Icon

Golden Construction

Stage: 5 Challenge Level: Challenge Level:1

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.

problem Icon

Always Two

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

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 Icon

Square Mean

Stage: 4 Challenge Level: Challenge Level:1

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

problem Icon

Partly Circles

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

What is the same and what is different about these circle questions? What connections can you make?

problem Icon

Interactive Number Patterns

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

How good are you at finding the formula for a number pattern ?

problem Icon

Target Six

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and explain why this is so. Find all the solutions of the equation.

problem Icon

Complex Sine

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Solve the equation sin z = 2 for complex z. You only need the formula you are given for sin z in terms of the exponential function, and to solve a quadratic equation and use the logarithmic function.

problem Icon

Pent

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

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 Icon

Xtra

Stage: 4 and 5 Challenge Level: Challenge Level:1

Find the sides of an equilateral triangle ABC where a trapezium BCPQ is drawn with BP=CQ=2 , PQ=1 and AP+AQ=sqrt7 . Note: there are 2 possible interpretations.

problem Icon

Put Out

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

After transferring balls back and forth between two bags the probability of selecting a green ball from bag 2 is 3/5. How many green balls were in bag 2 at the outset?

problem Icon

How Many Balls?

Stage: 5 Challenge Level: Challenge Level:1

A bag contains red and blue balls. You are told the probabilities of drawing certain combinations of balls. Find how many red and how many blue balls there are in the bag.

problem Icon

Kissing

Stage: 5 Challenge Level: Challenge Level:1

Two perpendicular lines are tangential to two identical circles that touch. How big is the circle that just fits between the two lines and the two circles and how would you construct it?

problem Icon

Cocked Hat

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

A new solution to a Tough Nut problem. Aleksander has drawn graphs for members of the family of functions given by the implicit equation (x^2 + 2ay -a^2)^2 = y^2(a^2 - x^2) corresponding to different. . . .

problem Icon

Power Quady

Stage: 4 Challenge Level: Challenge Level:1

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

problem Icon

Roots and Coefficients

Stage: 5 Challenge Level: Challenge Level:1

If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?

problem Icon

Pareq Calc

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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. . . .

problem Icon

Halving the Triangle

Stage: 5 Challenge Level: Challenge Level:1

Draw any triangle PQR. Find points A, B and C, one on each side of the triangle, such that the area of triangle ABC is a given fraction of the area of triangle PQR.

problem Icon

Proof Sorter - Quadratic Equation

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

problem Icon

Two Cubes

Stage: 4 Challenge Level: Challenge Level:1

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to. . . .