There are 37 NRICH Mathematical resources connected to Expanding and factorising quadratics, you may find related items under Algebraic expressions, equations and formulae.
Broad Topics > Algebraic expressions, equations and formulae > Expanding and factorising quadratics
Which armies can be arranged in hollow square fighting formations?
There are unexpected discoveries to be made about square numbers...
What is special about the difference between squares of numbers adjacent to multiples of three?
If you know the perimeter of a right angled triangle, what can you say about the area?
Surprising numerical patterns can be explained using algebra and diagrams...
Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?
Can you find the hidden factors which multiply together to produce each quadratic expression?
How do scores on dice and factors of polynomials relate to each other?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
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.
Can you produce convincing arguments that a selection of statements about numbers are true?
Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
This is a beautiful result involving a parabola and parallels.
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.
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
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?
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.
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?
Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
What do you get when you raise a quadratic to the power of a quadratic?
Can you prove our inequality holds for all values of x and y between 0 and 1?
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
If you plot these graphs they may look the same, but are they?
This polar equation is a quadratic. Plot the graph given by each factor to draw the flower.
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.
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.
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.
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.
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.
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.
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.
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.]