Track quiz
The table below has every idea in this track. Then thirteen questions, one per idea. Look at the table as often as you like.
| Idea | How it goes |
|---|---|
| Cutting down | (a − b)² = a² − 2ab + b². The corner is cut twice, so it comes back once. |
| Reading it back | Both ends must be whole squares. The middle must be twice their product. |
| One from another | a² − b² = (a − b)(a + b). The gap times the total gives the answer. |
| Three parts | (a+b+c)² = a²+b²+c²+2ab+2bc+2ca. Nine pieces: three squares, six matching crossings. |
| A cube | (a+b)³ = a³+3a²b+3ab²+b³. Two kinds arrive once. Two kinds arrive three at a time. |
| The pair | It must add to the middle number and multiply to the last number. |
| A number in front | The pair multiplies to the first number times the last. Split, group, take the bracket out. |
| Cancelling | A top and bottom may lose only a shared factor, never a term that is added on. |
| Adding two | Build one shared bottom, rewrite both tops, then add the tops. |
| Your own claim | One disagreement settles it. A few agreements do not. |
| Two letters | Square each part. The middle is twice their product, with the sign between the parts. |
| Two cubes | x³ − y³ = (x − y)(x² + xy + y²). x³ + y³ = (x + y)(x² − xy + y²). |
| Round numbers | a² = (a + b)(a − b) + b². Choose b to make a factor round. |
Which expression is a perfect square?
What is 45² − 35², without squaring either number?
How many terms does (a + b + c)² have after opening out?
A cube has edge a + b. Which kinds of block come three at a time?
Which pair splits the middle term of x² − 9x + 20?
For 4x² + 13x + 3, what must the pair multiply to?
Is (x + 4)/4 the same as x?
What is 1/x + 1/(x + 2)?
A claim says x² + 4 = 4x. At x = 2 both sides give 8. What does this show?
What is (x − 3)² after opening out?
What is (3m + 5n)² after opening out?
What is x³ − 8 as two brackets?
What is 59², using a² = (a + b)(a − b) + b² with b = 1?
Chapter 4 of your book is Exploring Algebraic Identities.
Bring the question you could not finish.
Next: I'm Up and Down, and Round and Round. A circle, its chords, and angles that stay equal.