Expanding and factorising: practice problems
Expanding and factorising are opposite processes. Learn the four identities so that you can use them in both directions.
Basic, Standard, Advanced: Grade 9 · Term 1
Math problem generator
The identities
For , treat as one letter: .
How to factorise
First take out any common factor, then look for an identity.
For , find two numbers with product and sum .
Substitution and tricks
With , .
Identities help with arithmetic too: and .
Worked examples
Expand.
Hint
Use .
Answer
Solution
Use the identity.
So
Simplify.
Hint
Expand each part, then combine like terms. Keep brackets around the part being subtracted.
Answer
Solution
Expand each part.
Remove the brackets and simplify.
Prove that for two consecutive integers, the square of the larger minus the square of the smaller equals their sum.
Hint
Write the numbers with letters, expand, and rearrange into the required form.
Answer
Write the integers as and . Then , which is their sum.
Solution
Plan: Write the integers as and compute .
Practice problems
Factorise.
Hint
Use .
Answer
Solution
Use the identity.
Expand.
Hint
Treat as a single letter and use an identity.
Answer
Solution
Let and use an identity.
Prove that for three consecutive integers, the square of the middle one minus 1 equals the product of the other two.
Hint
Write the numbers with letters, expand, and rearrange into the required form.
Answer
Write the integers as . The product of the outer two is , which is the square of the middle one minus 1.
Solution
Plan: Write the integers as and expand .