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There are **37** NRICH Mathematical resources connected to **Expanding and factorising quadratics**, you may find related items under Algebraic expressions, equations and formulae.

Problem
Primary curriculum
Secondary curriculum
### Hollow Squares

Which armies can be arranged in hollow square fighting formations?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square Number Surprises

There are unexpected discoveries to be made about square numbers...

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Difference of Two Squares

What is special about the difference between squares of numbers adjacent to multiples of three?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pythagoras Perimeters

If you know the perimeter of a right angled triangle, what can you say about the area?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Quadratic Patterns

Surprising numerical patterns can be explained using algebra and diagrams...

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Factorising with Multilink

Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Finding Factors

Can you find the hidden factors which multiply together to produce each quadratic expression?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Spinners

How do scores on dice and factors of polynomials relate to each other?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Perfectly Square

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pair Products

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Always Perfect

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

Age 14 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Telescoping Functions

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Number Rules - OK

Can you produce convincing arguments that a selection of statements about numbers are true?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Iff

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Parabella

This is a beautiful result involving a parabola and parallels.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Why 24?

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### What's Possible?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fair Shares?

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Plus Minus

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### How Old Am I?

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Always Two

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.

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### 2-digit Square

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Common Divisor

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Mega Quadratic Equations

What do you get when you raise a quadratic to the power of a quadratic?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Unit Interval

Can you prove our inequality holds for all values of x and y between 0 and 1?

Age 16 to 18

ShortChallenge Level

Problem
Primary curriculum
Secondary curriculum
### Geometric Parabola

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Spot the Difference

If you plot these graphs they may look the same, but are they?

Age 16 to 18

ShortChallenge Level

Problem
Primary curriculum
Secondary curriculum
### Polar Flower

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Multiplication Magic

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). The question asks you to explain the trick.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Poly Fibs

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fibonacci Factors

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Composite Notions

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Powerful Factors

Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Never Prime

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Novemberish

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Code to Zero

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Two Cubes

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 prove you have found all possible solutions.]

Age 14 to 16

Challenge Level