Equation or Identity?
Test values to see if a statement is always true or only sometimes true.
Equation
y = (x + 1)^2 - (x^2 + 0*x*1 + 1^2)
Graph
Table
| x | y |
|---|---|
| -5 | -10 |
| -4 | -8 |
| -3 | -6 |
| -2 | -4 |
| -1 | -2 |
| 0 | 0 |
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
| 4 | 8 |
| 5 | 10 |
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
- Find the k where y = 0 at x = 2 AND at x = −1 — true for EVERY x. That is an IDENTITY; any other k gives an equation that is true only sometimes.
- That special k is exactly 2: (x+a)² = x² + 2ax + a².
- A steeper square: (2x + a)². Set a = 3.
- Find the k that makes this true for EVERY x, not just one. The middle term is 2·(2x)·a, so k is 4 — anything else is an equation, not an identity.
Check yourself
Why does the graph staying flat at y=0 prove that (x+a)² = x²+2ax+a² is an identity?
What is the common mistake students make when expanding (x+3)²?
If a=2, what is the value of (x+2)² - (x²+4x+4) when x=5?
- Because it only works for x=0.
- Because the difference between LHS and RHS is always 0 for all x and a. — correct
- Because the graph is a straight line.
- Because we solved for x.
- Thinking it equals x² + 9 — correct
- Thinking it equals x² + 6
- Thinking it equals 2x + 6
- Thinking it equals x² + 3
- 10
- 25
- 0 — correct
- 4