The Difference of Squares Trick
See why (a+b)(a-b) always equals a² - b². No long multiplication needed.
Equation
y = (x + 1)*(x - 1) - (x^2 + 1*x - 9)
Graph
Table
| x | y |
|---|---|
| -5 | 13 |
| -4 | 12 |
| -3 | 11 |
| -2 | 10 |
| -1 | 9 |
| 0 | 8 |
| 1 | 7 |
| 2 | 6 |
| 3 | 5 |
| 4 | 4 |
| 5 | 3 |
What this lesson covers
What you do
You shape the function y = (x + a)*(x - a) - (x^2 + k*x - 9) and watch the graph answer.
Challenges to clear
- (x+a)(x−a): the middle terms CANCEL. Slide k and a until the difference is 0 at x = 2 and x = −1.
- So k = 0 (no middle term) and a = 3 (a² = 9): (x+3)(x−3) = x² − 9 exactly.
- In (x + a)(x − a) the two middle terms cancel exactly. Slide k to 0 — there is no x term at all.
- Now slide a until the difference is 0 at x = 2. The constant is −a², and 25 is 5 squared.
Check yourself
Why does the graph staying flat at y=0 prove the identity (x+a)(x-a) = x^2 - a^2?
When expanding (x+a)(x-a), why does the middle term disappear?
If a = 4, what is the value of (x+4)(x-4) when x = 5?
- Because y represents the difference between LHS and RHS; if y is always 0, LHS must equal RHS for all inputs. — correct
- Because the graph is a horizontal line, which means the equation has no solution.
- Because the parameter 'a' cancels out the variable 'x' completely.
- Because the function is undefined for any non-zero value of 'a'.
- The Outer and Inner products are equal in magnitude but opposite in sign (+ax and -ax), so they sum to zero. — correct
- Multiplying a positive term by a negative term always results in zero.
- The squares a^2 and x^2 absorb the middle terms during distribution.
- The identity only works when the middle term is explicitly removed by the user.
- 9 — correct
- 25
- 16
- 0