Year 9 · Unit 8 Ready for Year 10 · Lesson 3 of 6

Squaring a bracket, and a square with a corner cut away

(x + a)² has two strips as well as two squares, so it is not x² + a². And a square with a smaller square cut from its corner can be rebuilt as a rectangle, which is why x² − a² = (x + a)(x − a).

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. Work out 7² and (−7)².
    Answer49 and 49: (−7) × (−7) is positive.
  2. From lesson 2: expand (x + 3)(x − 8).
    Answerx² − 5x − 24: 3 + (−8) = −5 and 3 × (−8) = −24.
  3. What is the area of a square of side 12 cm?
    Answer144 cm², because 12 × 12 = 144.
Words used in this lesson
squared
multiplied by itself. For example: (x + 1)² means (x + 1)(x + 1).
perfect square
an expression that is something squared. For example: x² + 2x + 1 is (x + 1)².
difference of two squares
one square take away another. For example: x² − 25 = (x + 5)(x − 5).
identity
a statement that is true for every value of x. For example: (x + 1)² = x² + 2x + 1.
strip
a long thin rectangle in the grid. For example: the two strips of (x + 1)² are x and x.

Yusuf's shortcut

A square tile is x cm wide. A bigger tile is 4 cm wider each way, so its area is (x + 4)² square centimetres. Yusuf says that is x² + 16.

Predict first: is Yusuf right? Test his shortcut at x = 1 and at x = 10. If it is wrong, is it wrong by the same amount each time?

2
3
x² + 4x + 4the expansion
25the squared bracket at this x
12how far x² + a² is out

(x + 2)² = x² + 2x + 2x + 4 = x² + 4x + 4. At x = 3 it is 25, but x² + a² is 13: out by 12, the two strips.

Show me the steps
  1. x + a with a = 2 and x = 3: the cells are x², 2x, 2x and 4, so (x + 2)² = x² + 4x + 4. At x = 3: 5² = 25, but x² + a² = 9 + 4 = 13, which is 12 short: the two strips, 6 + 6.
  2. x − a with a = 4 and x = 9: the cells are x², −4x, −4x and +16, so (x − 4)² = x² − 8x + 16. At x = 9: 5² = 25, and 81 − 72 + 16 = 25.
  3. The middle is always two equal strips, so (x + a)² = x² + 2ax + a² and (x − a)² = x² − 2ax + a². The corner is +a² either way.

Squaring a bracket gives two squares and two equal strips.

(x + a)² = x² + 2ax + a² and (x − a)² = x² − 2ax + a². The 2ax is the two strips; the corner a² is positive in both, because (−a) × (−a) is positive.

For the tile, (x + 4)² = x² + 8x + 16. Yusuf leaves out the two strips, 8x, so he is 8 short at x = 1 (25 against 17) and 80 short at x = 10 (196 against 116). The bigger the tile, the bigger his mistake.

The difference of two squares

Multiply (x + a)(x − a) with the lesson 2 pattern: the numbers are a and −a, which add to 0, so the x terms cancel and the number term is −a². So x² − a² = (x + a)(x − a). The second figure shows the same fact as area: cut the corner, move one strip, and the leftover area is a rectangle.

7
2
45the area left, x² − a²
9 by 5the rectangle the pieces make, (x + a) by (x − a)

Square, corner cut: 7 by 7 with a 2 by 2 corner gone leaves 49 − 4 = 45.

Show me the steps
  1. x = 7, a = 2, square with the corner cut: 7 by 7 with a 2 by 2 corner gone leaves 49 − 4 = 45.
  2. Strip moved: the strip 2 wide and 5 long goes along the top. The shape is now 5 wide and 9 tall, and 9 × 5 = 45.
  3. x = 10, a = 3: 100 − 9 = 91, and the rectangle is 13 tall and 7 wide: 13 × 7 = 91.
  4. Always: the rectangle is x − a wide and x + a tall, so x² − a² = (x + a)(x − a).

A common mistake: using the difference of two squares on a sum. x² − 49 = (x + 7)(x − 7), but x² + 49 does not split this way: (x + 7)(x − 7) gives −49, not +49. The pattern needs a minus between two squares.

Worked examples

Expand (x − 5)².

  1. Cells: x², −5x, −5x and (−5)² = 25.
  2. x² − 10x + 25. Check at x = 8: 3² = 9 and 64 − 80 + 25 = 9.

Expand (x + 8)(x − 8).

  1. The numbers are 8 and −8: they add to 0, so no x term.
  2. x² − 64. Check at x = 10: 18 × 2 = 36 and 100 − 64 = 36.

Work out 52 × 48 in your head.

  1. 52 = 50 + 2 and 48 = 50 − 2, so 52 × 48 = 50² − 2².
  2. 2500 − 4 = 2496.

Your go. The first step is given. Expand (x − 7)².

  1. The two strips are −7x and −7x.
  2. Now find the corner yourself, collect, and check with a value.
Check the answer

Corner (−7)² = 49: x² − 14x + 49. Check at x = 10: 3² = 9 and 100 − 140 + 49 = 9.

Check before you go on

1. Which is (x + 5)² written without brackets?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. (x + 7)² = x² + bx + c. What is b?

3. Expand (x + 9)(x − 9). What is the number term?

Level 2 · Building

4. (x − 3)² = x² + px + q. What is p?

5. Use the difference of two squares to work out 31 × 29 without a calculator.

6. A square photo is x cm by x cm. A new frame makes one side 3 cm longer and the other 3 cm shorter. How many square centimetres smaller is the new rectangle than the square?

Level 3 · Stretch

7. (x + 4)² − (x − 4)² simplifies to kx. What is k?

8. Work out 67² − 33² without a calculator.

Something to think about

Work out 99² without a calculator by writing 99 as 100 − 1. Which is quicker: this, or 99 × 99 written out?

One way to think about it

(100 − 1)² = 100² − 2 × 100 × 1 + 1² = 10000 − 200 + 1 = 9801. The strips are −100 each and the corner is +1. The same idea gives 101² = 10000 + 200 + 1.

Exit ticket

9. Which is equal to (2x − 3)²?