Expand (x + 3)(x + 5)
See how the middle term is the sum of the constants, and the last term is their product.
Equation
y = (x + 3)*(x + 5) - (x^2 + 0*x + 3*5)
Graph
Table
| x | y |
|---|---|
| -5 | -40 |
| -4 | -32 |
| -3 | -24 |
| -2 | -16 |
| -1 | -8 |
| 0 | 0 |
| 1 | 8 |
| 2 | 16 |
| 3 | 24 |
| 4 | 32 |
| 5 | 40 |
What this lesson covers
What you do
You shape the function y = (x + a)*(x + b) - (x^2 + k*x + a*b) and watch the graph answer.
Challenges to clear
- Expand (x+3)(x+5): keep a = 3, b = 5 and slide k to the middle coefficient (the sum 3+5). Difference 0 at x = 2.
- And 0 at x = −1: middle term = sum, last term = product.
- Your own pair of brackets: (x + 2)(x + 7). Set a = 2 and b = 7.
- Now the middle coefficient. It is the SUM 2 + 7, while the last term is the PRODUCT 2 × 7. Slide k until the difference is 0 at x = 3.
Check yourself
Why does the graph staying flat at y=0 prove the identity?
What is the common mistake when expanding (x+a)(x+b)?
If a=4 and b=9, what is the constant term in the expansion?
- Because the graph is a straight line
- Because LHS - RHS = 0 means LHS equals RHS for all x — correct
- Because the parameters a and b are positive
- Because x^2 is always positive
- Forgetting to multiply the constants a and b — correct
- Adding a and b instead of multiplying them
- Squaring x twice
- Ignoring the x terms
- 13
- 36 — correct
- 49
- 4
Think about it
- In (x+3)(x+5) = x² + ?·x + 15, the middle number is the SUM of 3 and 5. What is it?