Look at the following figure. Justify the identity a² = (a + b) (a − b) + b² for yourself.

Step-by-step solution

Idea: Cutting a shape into pieces and rearranging them does not change the total area. Rearrange the a × a square into a rectangle plus a small square, then compare areas. (Algebraically, this is just a² − b² = (a + b)(a − b) with b² moved to the other side.)

1. Start with a square of side a. Its area is a².½ mark
2. Draw a line at height b from the bottom. The top part is a rectangle a wide and (a − b) tall. The bottom strip is a wide and b tall.½ mark
3. Split the bottom strip into a rectangle (a − b) × b and a small square b × b (area b²).½ mark
4. Turn the (a − b) × b rectangle on its side and place it against the right edge of the top part. It is b wide and (a − b) tall, so it fits exactly.½ mark
5. The top part has now become one rectangle of width a + b and height a − b, with area (a + b)(a − b). The only piece not used is the b × b square.½ mark
6. No area was lost or added, so a² = (a + b)(a − b) + b².
   Check by algebra: (a + b)(a − b) + b² = a² − b² + b² = a² ✓.½ mark

Check: a = 55, b = 5: (55 + 5)(55 − 5) + 5² = 60 × 50 + 25 = 3000 + 25 = 3025, and 55 × 55 = 3025 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Thinking the moved strip is a × b. Only the (a − b) × b part is moved; the b × b corner is left behind.
• Writing the new rectangle as a × (a + b). Its height is a − b, because the bottom strip of height b was removed.
• Forgetting to add b² when using the method for numbers, e.g. 55² = 60 × 50 = 3000 (wrong; it is 3025).

How this can come in the exam

Try one yourself

Use a² = (a + b)(a − b) + b² to find 38². (Choose b so that one factor is a round number.)

More questions like this

• 1. Try to evaluate the following using a suitable identity:
  (i) 35² (ii) 65² (iii) 85² (iv) 105²
  Do you observe any interesting pattern?
  2. Observe the two rows of figures below. They represent an algebraic identity. Try to identify it.
• Suppose 7x is split as 2x + 5x; can a similar rectangular arrangement be formed? Consider other possibilities and check.
• Algebra tiles can be used to represent products and find factors.
  1. Figure out the product of x + 2 and x + 3 using algebra tiles.
  2. Lay out algebra tiles for x² + 11x + 30 in such a way that you will see its factors.
• We have seen that (x + 3)(x + 4) = x² + 7x + 12.
  Also (x + 6)(x + 7) = x² + 13x + 42.
  Generalise the pattern to get an expression for (x + a) (x + b).
• Now consider the case where we have a rectangle of sidelengths 2x + 3 and 3x + 1, as shown in the figure. What can you say about its area (2x + 3) (3x + 1)?

All Algebraic identities questions · All maths questions