Mental Math: Square 98 Instantly
Use (a - b)^2 to solve 98^2 without long multiplication.
Equation
y = (100 - x)^2 - (10000 - 0*100*x + x^2)
Graph
Table
| x | y |
|---|---|
| 0 | 0 |
| 1 | -200 |
| 2 | -400 |
| 3 | -600 |
| 4 | -800 |
| 5 | -1000 |
| 6 | -1200 |
| 7 | -1400 |
| 8 | -1600 |
| 9 | -1800 |
| 10 | -2000 |
What this lesson covers
What you do
You shape the function y = (100 - x)^2 - (10000 - k*100*x + x^2) and watch the graph answer.
Challenges to clear
- 98 = 100 − 2: slide k until the shortcut (100−b)² = 10000 − k·100·b + b² is exact at b = 2.
- 95² too: zero at b = 5 — the same trick squares any number near 100 in your head.
- The shortcut works from ANY round base a, not just 100. Slide k until the difference is 0 at x = 3.
- Now use it on 57²: slide a to 60, because 57 = 60 − 3. That is 3600 − 360 + 9 = 3249, done in your head.
Check yourself
Why does the graph staying flat at y=0 prove the identity (100-a)² = 100² - 200a + a²?
Using the identity, what is the value of the middle term (-2*100*a) when a=2?
Why is it easier to calculate 98² using (100-2)² than multiplying 98*98 directly?
- Because y represents the difference between LHS and RHS; if the difference is always 0, they are equal. — correct
- Because the graph is a horizontal line, which means the numbers are large.
- Because 100 is a perfect square, so the identity only works for multiples of 10.
- Because the variable 'a' cancels out completely in the subtraction.
- -400 — correct
- -200
- -4
- 400
- Squaring 100 and 2 is mentally easier than multiplying two-digit numbers. — correct
- The identity reduces the number of digits in the final answer.
- It avoids the need for subtraction.
- It allows you to use a calculator.