Quadratic Expression Trick

The Trick

Have a friend think of a quadratic expression of the form ax² + bx + c.

Have her substitute 0, 1, and 2 for x and tell you the results (in order).

Then you will tell her what expression she was thinking of.

Now here is what you do to figure out what her expression was:

Determine the first and second differences of the three numbers (in order).
The first differences would be (2nd number - 1st number) and (3rd number - 2nd number)
The second difference would be the difference of the numbers in step two.

Now you can tell the person what expression they were thinking of:
The coefficient of x² is 1/2 times the last number.
The coefficient of x is the 1st number in the middle row (the first differnces) minus 1/2 the number in the bottom row (the 2nd differences)
The constant is the first number in the top row.

Here is an example:

Suppose your friend thinks of the quadratic expression 3x² - 5x + 2.

After she substitutes 0, 1, and 2 for x, she tells you the results: 2, 0, 4.
when x = 0, you get 2
when x = 1, you get 0
when x = 2, you get 4

Now here is what you do to figure out her expression:

Determine the first and second differences of the three numbers (in order).
2 0 4 (original 3 numbers)
-2 4 (first differences)
6 (second differences)

Now you can tell the person what expression they were thinking of:

The coefficient of x² is 1/2 times the last number, so a = 1/2 x 6 = 3.

The coefficient of x is the 1st number in the middle row - 1/2 the number in the bottom row, so b = (-2) - 1/2 (6) = -5

The constant is the first number in the top row, so c = 2.

Explanation of the Trick

Let f(x) = ax² + bx + c.
Then f(0) = c
Then f(1) = a + b + c
And f(2) = 4a + 2b + c

Now compute the first and second differences of these three numbers:
c a + b + c 4a + 2b + c
a + b 3a + b
2a

So, the coefficient of x² is 1/2 times the last number, which is 1/2 (2a) = a.

The coefficient of x is the 1st number in the middle row - 1/2 the number in the bottom row, which is (a + b) - 1/2 (2a) = b

The constant is the first number in the top row, which is c.