Spotting the Perfect Square
Recognise x^2 + 10x + 25 as (x + 5)^2 and factor it.
Equation
y = (x^2 + 10*x + 25) - (x + 1)^2
Graph
Table
| x | y |
|---|---|
| -5 | -16 |
| -4 | -8 |
| -3 | 0 |
| -2 | 8 |
| -1 | 16 |
| 0 | 24 |
| 1 | 32 |
| 2 | 40 |
| 3 | 48 |
| 4 | 56 |
| 5 | 64 |
What this lesson covers
What you do
You shape the function y = (x^2 + 10*x + 25) - (x + b)^2 and watch the graph answer.
Challenges to clear
- x² + 10x + 25 is a square in disguise: slide b until (x+b)² matches — 0 at x = 2. (Hint: half of 10.)
- Confirm at x = −1: it is exactly (x+5)².
- A perfect square starts with exactly ONE x². Slide a to 1.
- Now slide b until x² + 12x + 36 matches (x + b)² at x = 2. Halve the 12 — and check 36 really is that half squared.
Check yourself
Why does the graph staying flat at y=0 prove that x^2 + 2ax + a^2 = (x+a)^2?
Which of the following is a perfect square trinomial?
If a = 4, what is the value of the middle term 2ax when x = 3?
- Because the difference between the two sides is always zero, meaning they are equal for all x and a. — correct
- Because the graph is a straight line.
- Because x^2 is always positive.
- Because the slider only goes up to 5.
- x^2 + 6x + 9 — correct
- x^2 + 6x + 6
- x^2 + 9x + 3
- x^2 + 6x + 12
- 24 — correct
- 12
- 7
- 16
Think about it
- x² + 10x + 25: half of 10 is…?