The (a - b)² Identity Trap
Why (10 - 3)² is NOT 100 - 9. Watch the middle term appear.
Equation
y = (3*x - 1)^2 - (9*x^2 - 0*x*1 + 1^2)
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 |
Goals
- A steeper difference: (3x − a)². Set a = 4.
- The middle term is 2·(3x)·a = 6xa, and it is NEGATIVE. Slide k until the difference is 0 at x = 2.
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
- (x−a)² hides a middle term: slide k until y = 0 at x = 2. ((10−3)² is 100 − 60 + 9, NOT 100 − 9!)
- Identity: 0 at x = −1 too — the middle term is always −2ax.
- A steeper difference: (3x − a)². Set a = 4.
- The middle term is 2·(3x)·a = 6xa, and it is NEGATIVE. Slide k until the difference is 0 at x = 2.
Check yourself
Why does the graph staying flat at y=0 prove the identity?
What is the common mistake this lesson helps avoid?
If a=3 and x=5, what is the value of (x - a)²?
- Because LHS - RHS = 0 means LHS = RHS for every x and a. — correct
- Because the graph is a straight line.
- Because x and a cancel each other out.
- Because squaring always results in zero.
- Thinking (x - a)² = x² - a² — correct
- Thinking (x - a)² = x² + a²
- Thinking (x - a)² = x² - 2a
- Thinking (x - a)² = 2x - 2a
- 4
- 16 — correct
- 25
- 9