Binomial formulas with squares

What is it about?

The three binomial formulas are shortcuts for expanding brackets:

\begin{aligned} (a + b)^2 &= a^2 + 2ab + b^2 \\ (a - b)^2 &= a^2 - 2ab + b^2 \\ (a + b)(a - b) &= a^2 - b^2 \end{aligned}

You can see them as areas: a square with side length $a + b$ consists of the square $a^2$ , the square $b^2$ and two rectangles with area $a \cdot b$ . These two rectangles are exactly what is forgotten in the common mistake $(a + b)^2 = a^2 + b^2$ .

The formulas hold for all numbers – the areas are only an illustration for positive lengths (with $a > b$ for the 2nd and 3rd formula).

Try it

Press Split apart and watch which parts the large square falls into. How many rectangles $a \cdot b$ are there?
Choose the second formula. Why do you have to add $b^2$ again, even though something is being subtracted? 2nd formula
Choose the third formula and let the strip be moved. Which rectangle appears? 3rd formula
Change $a$ and $b$ and check the calculation with numbers under Results.

Exercises

Exercise 1: Clever mental arithmetic

Use a binomial formula to calculate in your head: $53^2$ , $49^2$ and $21 \cdot 19$ .

Show solution

$53^2 = (50 + 3)^2 = 2500 + 300 + 9 = 2809$
$49^2 = (50 - 1)^2 = 2500 - 100 + 1 = 2401$
$21 \cdot 19 = (20 + 1)(20 - 1) = 400 - 1 = 399$

Exercise 2: Expanding

Expand: $(x + 4)^2$ , $(2x - 3)^2$ and $(5 + y)(5 - y)$ .

Show solution

$(x + 4)^2 = x^2 + 8x + 16$
$(2x - 3)^2 = 4x^2 - 12x + 9$ – careful: $(2x)^2 = 4x^2$ and $2 \cdot 2x \cdot 3 = 12x$ .
$(5 + y)(5 - y) = 25 - y^2$

Exercise 3: Find the mistake

Tom calculates $(a + b)^2 = a^2 + b^2$ . Check with $a = 5$ and $b = 2$ and use the simulation to explain what Tom forgot.

Show solution

$(5 + 2)^2 = 49$ , but $5^2 + 2^2 = 29$ . The missing $20 = 2 \cdot 5 \cdot 2$ are the two rectangles $a \cdot b$ . Show in the simulation

Exercise 4: Backwards – factorising

Write as a product: $x^2 + 10x + 25$ , $49 - y^2$ and $9a^2 - 12ab + 4b^2$ .

Show solution

$x^2 + 10x + 25 = (x + 5)^2$
$49 - y^2 = (7 + y)(7 - y)$
$9a^2 - 12ab + 4b^2 = (3a - 2b)^2$

Ideas for the classroom

Use: About grade 7/8 (terms with brackets, binomial formulas). The area model works as an introduction and as a justification of why the middle term $2ab$ appears.
Animation: The button below the drawing area splits the figure apart or puts it back together. For the third formula a strip is rotated and moved – a visual justification of $a^2 - b^2$ .
Common stumbling blocks:
- $(a + b)^2 = a^2 + b^2$ (“the square is distributed over the terms”).
- In $(a - b)^2$ the $+b^2$ feels wrong. The simulation shows that $b^2$ is subtracted twice when the two strips are removed.
- $(2x)^2 = 2x^2$ instead of $4x^2$ .
Limits of the model: Areas need positive lengths, and for the 2nd and 3rd formula also $a > b$ . If $a \le b$ , the simulation shows no area model for these formulas, but the calculation with numbers under Results is still correct, e.g. $(3 - 5)^2 = 4 = 9 - 30 + 25$ . The formulas also hold for negative numbers – you can let students check this with numerical examples.