Finding factors
Can you find the hidden factors which multiply together to produce each quadratic expression?
Problem
The interactivity below is similar to the one used in Missing Multipliers so you may wish to work on that first.
In the multiplication grid below, the headings and the answers have been hidden. Each of the headings is an expression of the form $x + a$ where $a$ is an integer between $-5$ and $5$. By revealing some of the answers, can you work out what each heading must be?
Drag the green and purple labels onto the headers to make the correct expressions.
What is the smallest number of answers you need to reveal in order to work out the missing headers?
Once you've developed strategies for finding the factors in the interactivity above, you might like to have a go at solving some bigger grids which include coefficients of $x$ greater than 1 in the row and column headings:
Six by six grid
Eight by eight grid
Ten by ten grid
You may also be interested in the other problems in our Working backwards, leaping forwards Feature.
Getting Started
What does the constant term of the quadratic expression tell you about the headers?
What does the coefficient of x tell you about the headers?
Teachers' Resources
Why do this problem?
This problem could be used as an introduction to factorising quadratic expressions, or to develop students' fluency in this skill. The 'hook' of an interactive environment draws students in, encouraging them to be resilient as they strive to complete the challenge.
Possible approach
Students will need to be able to expand pairs of brackets of the form $(x \pm a)(x \pm b)$ before embarking on this problem - Pair Products provides a nice opportunity to practise this.
Once they are confident at tackling these examples, they could try one of the larger grids mentioned in the problem, where the quadratics are of the form $ax^2+bx+c$.
If individual computers are not available, this could be done as a whole class activity where a few cells are revealed and students are invited to work out as many factors as they can, before requesting the revealing of further cells. Alternatively, students could create their own grids by choosing pairs of brackets for the two columns, multiplying them out, and then revealing certain cells to a partner.
Key questions
What does the constant term of the quadratic expression tell you about the numbers in the headers?
Possible support
Factorising with Multilink offers a visual representation of the process of factorising quadratics, which some students may find helpful.
Possible extension
How Old Am I? invites students to solve a series of problems that can be modelled with quadratic equations, leading to some generalisations.
Submit a solution
What we like to see
We have written an article on what we are looking for when we decide which solutions to publish.
Explain your methods clearly for other school students to read.
Give reasons and convincing arguments or proofs where you can.
You can share your work with us by typing into the form or uploading a file (must be less than 10MB.)
What can I upload?
- Word documents
- PDF files
- Images - any format
- Screenshots
- Photos of written work
If your work is on another app
You can also send us a link to your work on:
- Youtube/Vimeo
- Google Docs
- Sharepoint/Office 365
- Anywhere sensible on the internet!
How we use your information
By uploading your solution here you are giving us permission to publish your work, either as a whole or edited extracts, on the NRICH website and in associated educational materials for students and teachers, at our discretion.
For our full terms of use regarding submitted solutions and user contributions please see https://nrich.maths.org/terms
Your information (first name, school name etc) is optional. If supplied, it may be used for attribution purposes only if your work is published on the website. Data that you supply on this form (e.g. age if supplied, location if supplied) may also be anonymised and used for evaluation and reporting.
For more information about the University's policies and procedures on handling personal information, and your rights under data protection legislation, please see https://www.information-compliance.admin.cam.ac.uk/data-protection/general-data.