The Trap of Squaring a Sum
Why (a + b)² is NOT a² + b². Watch the 'middle term' appear.
Equation
y = (x + 3)^2 - (x^2 + 0*x*3 + 3^2)
Graph
Table
| x | y |
|---|---|
| -5 | -30 |
| -4 | -24 |
| -3 | -18 |
| -2 | -12 |
| -1 | -6 |
| 0 | 0 |
| 1 | 6 |
| 2 | 12 |
| 3 | 18 |
| 4 | 24 |
| 5 | 30 |
What this lesson covers
What you do
You shape the function y = (x + a)^2 - (x^2 + k*x*a + a^2) and watch the graph answer.
Challenges to clear
- (a+b)² is NOT a² + b²: starting from k = 0 (the trap!), slide k until the difference is 0 at x = 2 — the middle term appears.
- 0 at x = −1 as well: the expansion is x² + 2ax + a², for every x.
- Steeper again: (4x + a)². Set a = 3.
- Starting from k = 0 — the classic trap — slide k until the middle term appears and the difference is 0 at x = 2. It is 2·(4x)·a = 8xa.
Check yourself
Why does the graph staying flat at y=0 prove that (x+a)² = x²+2ax+a²?
A student claims that (x + 5)² = x² + 25. What is the error in their reasoning?
If a = 4, what is the value of the middle term (2*x*a) when x = 3?
- Because y represents the difference between LHS and RHS; if the difference is always 0, they are equal. — correct
- Because the graph is a straight line.
- Because x and a are both positive numbers.
- Because squaring always results in a positive number.
- They forgot the middle term: 2*x*5 = 10x — correct
- They squared the 5 incorrectly.
- They should have subtracted 25 instead of adding it.
- There is no error; the identity is correct.
- 24 — correct
- 12
- 7
- 16