Square of a sum
A square garden has sides of 4 metres. Each side is made 3 metres longer, so the new garden has sides of 4 + 3 = 7 metres. To find the new area, the garden can be cut into four pieces.
Multiplying a sum of two terms by itself gives the square of a sum. The answer has 3 terms: the square of the first, twice the product of the two, and the square of the second. The formula is (a + b)² = (a + b)(a + b) = a² + 2ab + b².
In the figure: Here a = 4 and b = 3. The big square has side 4 + 3 = 7, so its area is 7² = 49. It has 4 pieces: a 4 × 4 = 16 square, two rectangles of 4 × 3 = 12 each, and a small 3 × 3 = 9 square. So 16 + 12 + 12 + 9 = 49. The two rectangles together give 2 × 12 = 24.
Rule 1: Use the formula for the square of a sum — Write the square of the first term, then twice the product of the two terms, and finally the square of the second term. All three terms carry a + sign.
- Easy: (x + 5)² = x² + 2 × x × 5 + 5² = x² + 10x + 25.
- Medium: (3m + 2)² = (3m)² + 2 × 3m × 2 + 2² = 9m² + 12m + 4.
- Exam type: (4p + 3q)² = (4p)² + 2 × 4p × 3q + (3q)² = 16p² + 24pq + 9q².
Rule 2: Name the first term and the second term first — Treat the two terms inside the bracket as a and b. Then work out a², 2ab and b² separately and write them down. If a term has a coefficient, put it in a bracket before squaring.
- Easy: (y + 1)² = y² + 2y + 1. Here the first term is y and the second term is 1.
- Medium: (2a + 7)² = 4a² + 28a + 49. Here the middle term is 2 × 2a × 7 = 28a.
- Exam type: (5x + 2y)² = 25x² + 20xy + 4y². Here the middle term is 2 × 5x × 2y = 20xy.
Rule 3: Test the formula with numbers — Put numbers in place of a and b and compare both sides. If both sides give the same value, the formula has been used correctly.
- Easy: (1 + 1)² = 2² = 4 and 1² + 2 × 1 × 1 + 1² = 1 + 2 + 1 = 4.
- Medium: (6 + 4)² = 10² = 100 and 6² + 2 × 6 × 4 + 4² = 36 + 48 + 16 = 100.
- Exam type: (5 + 2)² = 7² = 49, but 5² + 2² = 25 + 4 = 29. So (a + b)² = a² + b² is wrong; the middle term 2ab is needed.
Square of a sum
In the app you can move this figure with your finger.
Open in the appThe answer for (a + b)² has 3 terms. The middle term 2ab is the combined area of the two rectangles.
Writing (a + b)² = a² + b². This forgets the middle term. In the figure 16 + 9 = 25, but the square is 49; the two rectangles, 24, are left out.
Say "square of the first, square of the last, and twice the product in between" and write the three terms one after another.
Spot the difference
| Point | (a + b)² | (a − b)² |
|---|---|---|
| Middle term | 2ab | −2ab |
| Last term | b² | b² |
| Value for a = 5, b = 2 | 49 | 9 |
| In the figure | two rectangles are added to a square | strips are removed from a square |
Practice questions
- 1Expand (x + 6)².
- 2Expand (m + 9)².
- 3Find the value of (3 + 8)² with the formula and check it against 11².
- 4Expand (5k + 3)².
- 5Expand (7a + 2b)².
- 6Rohan wrote the answer of (x + 3)² as x² + 9. Point out his mistake and write the correct answer.
- 7Simplify (2p + 3)² − 12p.
- 8A square has side (3m + 1) cm. Write its area in terms of m and find its value for m = 3.
Show answers (try them yourself first)
- 1(x + 6)² = x² + 12x + 36. 2) (m + 9)² = m² + 18m + 81. 3) 3² + 2 × 3 × 8 + 8² = 9 + 48 + 64 = 121 and 11² = 121; both are equal.
- 4(5k + 3)² = 25k² + 30k + 9. 5) (7a + 2b)² = 49a² + 28ab + 4b². 6) The middle term was forgotten. 2 × x × 3 = 6x, so the correct answer is (x + 3)² = x² + 6x + 9.
- 7(2p + 3)² − 12p = 4p² + 12p + 9 − 12p = 4p² + 9.
- 8(3m + 1)² = 9m² + 6m + 1. For m = 3 the side is 3 × 3 + 1 = 10 cm and the area is 10² = 100 square cm. By the formula, 9 × 3² + 6 × 3 + 1 = 81 + 18 + 1 = 100.
Square of a difference
A farmer sold a strip 2 metres wide along two neighbouring sides of a square plot with sides of 6 metres. The land that is left is a square with side 6 − 2 = 4 metres.
Multiplying a difference of two terms by itself gives the square of a difference. Here the middle term is negative and the last term is positive. The formula is (a − b)² = (a − b)(a − b) = a² − 2ab + b².
In the figure: Here a = 6 and b = 2. The big square is 6 × 6 = 36 and has 4 parts: the remaining square of 16, two rectangles of 8 and 8, and the corner square of 4. So 16 + 8 + 8 + 4 = 36. Taking away the strips, 12 + 12 = 24, removes the corner 4 twice, so 36 − 24 + 4 = 16 = 4².
Rule 1: Use the formula for the square of a difference — Write the square of the first term, then the middle term with a minus sign, and finally the square of the second term with a plus sign.
- Easy: (x − 4)² = x² − 2 × x × 4 + 4² = x² − 8x + 16.
- Medium: (5m − 2)² = (5m)² − 2 × 5m × 2 + 2² = 25m² − 20m + 4.
- Exam type: (3p − 4q)² = (3p)² − 2 × 3p × 4q + (4q)² = 9p² − 24pq + 16q².
Rule 2: Test the formula with numbers — Both sides must give the same value. Numbers also show at once that (a − b)² is not a² − b².
- Easy: (5 − 2)² = 3² = 9 and 5² − 2 × 5 × 2 + 2² = 25 − 20 + 4 = 9.
- Medium: (10 − 3)² = 7² = 49 and 10² − 2 × 10 × 3 + 3² = 100 − 60 + 9 = 49.
- Exam type: (8 − 3)² = 5² = 25, but 8² − 3² = 64 − 9 = 55. So (a − b)² = a² − b² is wrong.
Rule 3: Reversing the order inside the bracket gives the same square — The squares of a − b and b − a are equal, because negative × negative = positive.
- Easy: (x − 5)² = (5 − x)² = x² − 10x + 25.
- Medium: (3 − 7)² = (−4)² = 16 and (7 − 3)² = 4² = 16.
- Exam type: (2 − 3y)² = (3y − 2)² = 9y² − 12y + 4.
Square of a difference
In the app you can move this figure with your finger.
Open in the appIn (a − b)² the middle term is −2ab, but the last term b² is positive, because negative × negative = positive.
Writing (x − 3)² = x² − 6x − 9. The last term is not −9; it is 3² = 9 and positive.
Recall the formula for the square of a sum and just flip the middle sign; the other two terms stay the same.
Spot the difference
| Point | (a − b)² | a² − b² |
|---|---|---|
| Meaning | square of a difference | difference of squares |
| Number of terms | 3 | 2 |
| Expansion | a² − 2ab + b² | a² − b² |
| Value for a = 9, b = 4 | 25 | 65 |
Practice questions
- 1Expand (y − 7)².
- 2Expand (m − 10)².
- 3Find the value of (9 − 4)² with the formula and check it against 5².
- 4Expand (8k − 3)².
- 5Expand (9a − 4b)².
- 6A boy wrote the answer of (p − 8)² as p² − 64. Point out his mistake and write the correct answer.
- 7Simplify (3x − 2)² + 12x.
- 8Two numbers a and b differ by 5 and their product is 14. Find the value of a² + b².
Show answers (try them yourself first)
- 1(y − 7)² = y² − 14y + 49. 2) (m − 10)² = m² − 20m + 100. 3) 9² − 2 × 9 × 4 + 4² = 81 − 72 + 16 = 25 and 5² = 25; both are equal.
- 4(8k − 3)² = 64k² − 48k + 9. 5) (9a − 4b)² = 81a² − 72ab + 16b². 6) The middle term was forgotten. 2 × p × 8 = 16p, so the correct answer is (p − 8)² = p² − 16p + 64.
- 7(3x − 2)² + 12x = 9x² − 12x + 4 + 12x = 9x² + 4.
- 8(a − b)² = a² − 2ab + b², so a² + b² = (a − b)² + 2ab. Now 5² + 2 × 14 = 25 + 28 = 53.
Product of a sum and a difference
A square piece of cloth has sides of 7 metres. A square with sides of 3 metres is cut from one corner, so the rest looks like a bent strip. If it is cut and joined again it forms a rectangle, and we want its length and width.
Multiplying the sum and the difference of the same two terms gives a difference of two squares. The middle terms cancel each other. The formula is (a + b)(a − b) = a² − b².
In the figure: Here a = 7 and b = 3. The square with side 7 has area 49 and the cut-out square with side 3 has area 9, so what is left is 49 − 9 = 40. The same piece fits in a rectangle that is 7 + 3 = 10 long and 7 − 3 = 4 wide; 10 × 4 = 40.
Rule 1: Use the formula for a sum times a difference — Write the square of the first term and subtract the square of the second term. Only 2 terms are left in the answer.
- Easy: (x + 6)(x − 6) = x² − 36.
- Medium: (6m + 5)(6m − 5) = 36m² − 25.
- Exam type: (7p + 2q)(7p − 2q) = 49p² − 4q².
Rule 2: Both brackets must hold the same terms with only the sign flipped — Only then can this formula be used. It does not matter if the order of the terms in a bracket is changed.
- Easy: The formula works on (y + 9)(y − 9). It does not work on (y + 9)(y − 5), because (y + 9)(y − 5) = y² − 81 is wrong.
- Medium: (8 − 3z)(8 + 3z) = 64 − 9z².
- Exam type: (2a − 5b)(5b + 2a) = 4a² − 25b².
Rule 3: Test the formula with numbers — Both sides must give the same value. Numbers also show that this formula differs from (a − b)².
- Easy: (8 + 3)(8 − 3) = 11 × 5 = 55 and 8² − 3² = 64 − 9 = 55.
- Medium: (12 + 5)(12 − 5) = 17 × 7 = 119 and 12² − 5² = 144 − 25 = 119.
- Exam type: (9 + 4)(9 − 4) = 13 × 5 = 65, but (9 − 4)² = 5² = 25. So (a + b)(a − b) = (a − b)² is wrong.
Product of a sum and a difference
In the app you can move this figure with your finger.
Open in the appThe middle terms −ab and +ab cancel each other: (a + b)(a − b) = a² − ab + ab − b² = a² − b².
Writing (x + 5)(x − 5) = x² + 25. A minus sign stays between the squares: (x + 5)(x − 5) = x² − 25.
Square the first term, square the second term and keep a minus between them; nothing has to be written in the middle.
Spot the difference
| Point | (a + b)(a − b) | (a − b)² |
|---|---|---|
| Number of terms | 2 | 3 |
| Expansion | a² − b² | a² − 2ab + b² |
| Middle term | cancels out | −2ab |
| Value for a = 9, b = 4 | 65 | 25 |
Practice questions
- 1Expand (m + 8)(m − 8).
- 2Expand (k − 11)(k + 11).
- 3Find the value of (15 + 5)(15 − 5) with the formula and check it against 20 × 10.
- 4Expand (6x + 7)(6x − 7).
- 5Expand (9a − 2b)(9a + 2b).
- 6Simplify (p + 3)(p − 3) + 9.
- 7Simplify (x + 4)(x − 4) − (x − 2)(x + 2).
- 8The sum of two numbers is 14 and their difference is 6. What is the difference of their squares?
Show answers (try them yourself first)
- 1(m + 8)(m − 8) = m² − 64. 2) (k − 11)(k + 11) = k² − 121. 3) 15² − 5² = 225 − 25 = 200 and 20 × 10 = 200; both are equal.
- 4(6x + 7)(6x − 7) = 36x² − 49. 5) (9a − 2b)(9a + 2b) = 81a² − 4b². 6) (p + 3)(p − 3) + 9 = p² − 9 + 9 = p².
- 7(x + 4)(x − 4) − (x − 2)(x + 2) = x² − 16 − x² + 4 = −12.
- 8a² − b² = (a + b)(a − b), so 14 × 6 = 84.
Finding values of numbers with formulas
A shopkeeper has to sell 63 articles at 63 rupees each. To work out 63 × 63 in the head he splits 63 into 60 + 3, because 60 is an easy round number.
When a number is written as a nearby round number plus or minus a small amount, the formulas for squares and products can be used. A big multiplication becomes a mental one. For squares use (a + b)² or (a − b)², and for the product of two numbers at equal distances from a round number use (a + b)(a − b).
In the figure: 63 is written as 60 + 3. Then 60² + 2 × 60 × 3 + 3² = 3600 + 360 + 9 = 3969. So 63² = 3969.
Rule 1: If the number is a little above a round number, use the formula for a sum — Write the number as a round number plus a small amount and square it with (a + b)².
- Easy: 52² = (50 + 2)² = 50² + 2 × 50 × 2 + 2² = 2500 + 200 + 4 = 2704.
- Medium: 71² = (70 + 1)² = 70² + 2 × 70 × 1 + 1² = 4900 + 140 + 1 = 5041.
- Exam type: 305² = (300 + 5)² = 300² + 2 × 300 × 5 + 5² = 90000 + 3000 + 25 = 93025.
Rule 2: If the number is a little below a round number, use the formula for a difference — Write the number as a round number minus a small amount and square it with (a − b)². The last term is positive.
- Easy: 49² = (50 − 1)² = 50² − 2 × 50 × 1 + 1² = 2500 − 100 + 1 = 2401.
- Medium: 88² = (90 − 2)² = 90² − 2 × 90 × 2 + 2² = 8100 − 360 + 4 = 7744.
- Exam type: 197² = (200 − 3)² = 200² − 2 × 200 × 3 + 3² = 40000 − 1200 + 9 = 38809.
In the figure: In 47 × 53 we have 47 = 50 − 3 and 53 = 50 + 3. So the product is 50² − 3² = 2500 − 9 = 2491.
Rule 3: Product of numbers at equal distances on both sides of a round number — Find the round number between the two numbers and use (a + b)(a − b) = a² − b².
- Easy: 48 × 52 = (50 − 2)(50 + 2) = 50² − 2² = 2500 − 4 = 2496.
- Medium: 76 × 84 = (80 − 4)(80 + 4) = 80² − 4² = 6400 − 16 = 6384.
- Exam type: 197 × 203 = (200 − 3)(200 + 3) = 200² − 3² = 40000 − 9 = 39991.
Rule 4: Check the last digit of the answer — The last digit of a square is the last digit of the square of the number's last digit. The last digit of a product is the last digit of the product of the two last digits.
- Easy: For 52², 2² = 4, and the answer 2704 ends in 4.
- Medium: For 88², 8² = 64, so the last digit is 4; the answer 7744 ends in 4.
- Exam type: 59 × 61 = (60 − 1)(60 + 1) = 60² − 1² = 3600 − 1 = 3599. The last digit is 9 × 1 = 9 and the answer ends in 9.
Finding values of numbers with formulas
In the app you can move this figure with your finger.
Open in the appChoose a round number whose square can be found in the head (60² = 3600) and keep the difference small.
Writing only 60² + 3² = 3600 + 9 = 3609 for 63². The middle term 2 × 60 × 3 = 360 was forgotten, so the answer is wrong; the correct answer is 3969.
Square of a number ending in 5: 35² = (30 + 5)² = 900 + 300 + 25 = 1225. Here 3 × 4 = 12, and writing 25 after it gives 1225.
Spot the difference
| Point | For a square | For a product |
|---|---|---|
| When to use | the number is near a round number | the numbers are at equal distances on both sides of a round number |
| Formula | (a + b)² or (a − b)² | (a + b)(a − b) |
| Terms in the expansion | 3 | 2 |
| Example | 88² = (90 − 2)² = 7744 | 48 × 52 = (50 − 2)(50 + 2) = 2496 |
Practice questions
- 1Use a formula to find the value of 41².
- 2Use a formula to find the value of 99².
- 3Use a formula to find the value of 28 × 32.
- 4Use a formula to find the value of 104².
- 5Use a formula to find the value of 67 × 73.
- 6Use a formula to find the value of 58².
- 7Use a formula to find the value of 996 × 1004.
- 8Use a formula to find the value of 75² and check the last digit.
Show answers (try them yourself first)
- 141² = (40 + 1)² = 1600 + 80 + 1 = 1681. 2) 99² = (100 − 1)² = 10000 − 200 + 1 = 9801. 3) 28 × 32 = (30 − 2)(30 + 2) = 900 − 4 = 896.
- 4104² = (100 + 4)² = 10000 + 800 + 16 = 10816. 5) 67 × 73 = (70 − 3)(70 + 3) = 4900 − 9 = 4891. 6) 58² = (60 − 2)² = 3600 − 240 + 4 = 3364.
- 7996 × 1004 = (1000 − 4)(1000 + 4) = 1000000 − 16 = 999984.
- 875² = (70 + 5)² = 4900 + 700 + 25 = 5625. The last digit comes from 5² = 25, which ends in 5, and the answer ends in 5.
Factorisation
A big rectangle is made of two smaller rectangles that have the same height, so its area is written as the sum of their areas. Because the heights match, the two pieces can be joined into one rectangle. Its width is the sum of the two widths.
Writing an expression as a product of two or more expressions is called factorisation. It is the opposite of multiplying; multiplying the factors gives back the original expression.
In the figure: The monomial 18pq² has been broken into factors. 18 = 2 × 3 × 3 and q² = q × q. So 18pq² = 2 × 3 × 3 × p × q × q.
Rule 1: Factors of a monomial: separate the number, the letters and the powers — Break the number into prime factors and write each power of a letter as a product.
- Easy: 12ab = 2 × 2 × 3 × a × b.
- Medium: 20x²y = 2 × 2 × 5 × x × x × y.
- Exam type: 30p²q = 2 × 3 × 5 × p × p × q.
In the figure: Both rectangles have height 3a. One has width 2b and area 3a × 2b = 6ab. The other has width 3a and area 3a × 3a = 9a². Joined, 6ab + 9a² = 3a(2b + 3a).
Rule 2: Take the common factor outside the bracket — Find the greatest common number factor and the common letters of all the terms. Write it outside the bracket, and inside the bracket write what is left after dividing each term by it.
- Easy: 6ab + 9a² = 3a(2b + 3a).
- Medium: 15m²n + 10mn² = 5mn(3m + 2n).
- Exam type: 18p²q + 24pq² = 6pq(3p + 4q). Here 6pq × 3p = 18p²q and 6pq × 4q = 24pq².
Rule 3: Check the factors by multiplying — Multiply the factor outside the bracket with every term inside. If the original expression comes back, the factors are right.
- Easy: 5(x + 4) = 5x + 20, so 5x + 20 = 5(x + 4).
- Medium: 7y(y − 3) = 7y² − 21y, so 7y² − 21y = 7y(y − 3).
- Exam type: 12x² + 18x = 3x(4x + 5) is wrong, because multiplying gives 12x² + 15x. The correct answer is 12x² + 18x = 3x(4x + 6) = 6x(2x + 3); the first is right but incomplete, the second is complete.
In the figure: In 16p² − 49q² both terms are squares: (4p)² and (7q)². So 16p² − 49q² = (4p + 7q)(4p − 7q). Check: multiplying gives 16p² − 28pq + 28pq − 49q². The middle terms cancel and 16p² − 49q² comes back.
Rule 4: Factors of a difference of two squares — a² − b² = (a + b)(a − b). Check that both terms are squares with a minus between them. If there is a common factor, take it out first and then use this formula.
- Easy: x² − 49 = (x + 7)(x − 7).
- Medium: 25m² − 9n² = (5m + 3n)(5m − 3n).
- Exam type: 3x² − 48 = 3(x² − 16) = 3(x + 4)(x − 4).
Factorisation
In the app you can move this figure with your finger.
Open in the appFactorisation
In the app you can move this figure with your finger.
Open in the appTake out the greatest common factor and check the factors by multiplying. For the terms 6ab and 9a² the greatest common factor is 3a.
Forgetting the 1 when a whole term is common: 5m + 5 = 5(m) is wrong; the right answer is 5m + 5 = 5(m + 1).
First ask, "Is anything common to all the terms?" If so, take it out. Then ask, "Are there two squares with a minus between them?" If so, use the formula for a² − b².
Spot the difference
| Point | Common factor | Difference of two squares |
|---|---|---|
| When it is used | something is common to all terms | two squares with a minus between them |
| Example | 6ab + 9a² | 16p² − 49q² |
| Factors | 3a(2b + 3a) | (4p + 7q)(4p − 7q) |
| Shape of the brackets | one bracket | two brackets, one with + and one with − |
Practice questions
- 1Write the factors of 14xy.
- 2Factorise 9m + 12.
- 3Factorise y² − 64.
- 4Factorise 21a²b + 35ab².
- 5Factorise 49x² − 25y².
- 6A student wrote the answer for 10p² + 15p as 5(2p² + 3p). Why is it incomplete? Write the complete answer.
- 7Factorise 2x² − 50 completely.
- 8Factorise (x + 5)² − 25.
Show answers (try them yourself first)
- 114xy = 2 × 7 × x × y. 2) 9m + 12 = 3(3m + 4). 3) y² − 64 = (y + 8)(y − 8).
- 421a²b + 35ab² = 7ab(3a + 5b). 5) 49x² − 25y² = (7x + 5y)(7x − 5y). 6) Inside the bracket 2p² + 3p still has p in common; the complete answer is 10p² + 15p = 5p(2p + 3).
- 72x² − 50 = 2(x² − 25) = 2(x + 5)(x − 5).
- 8Use a² − b²: (x + 5)² − 25 = (x + 5 + 5)(x + 5 − 5) = (x + 10)(x). So (x + 5)² − 25 = x² + 10x = x(x + 10).