Think of numbers a and b where a and b do not represent lengths of line segments. What if a and b are negative numbers? Let us check for some negative numbers and see if this equation still works.
- (i) Let a = −2 and b = −3.
- (ii) Now suppose a and b are rational numbers, say a = −23 and b = 34.
- Why it works for all numbers But we are still not sure if it is true for all numbers. To verify this, let us investigate further using the distributive property of numbers:
Step-by-step solution
To find: Whether both sides are still equal for these values
Idea: The square picture only proves (a + b)2 = a2 + 2ab + b2 for lengths, which are positive. To test other numbers, work out the left side and the right side separately and compare. Finally, the distributive property proves it for all numbers.
(i) Let a = −2 and b = −3.
- Left side: a + b = −2 + (−3) = −5, so (a + b)2 = (−5)2 = 25.½ mark
- Right side: a2 = (−2)2 = 4, b2 = (−3)2 = 9, 2ab = 2 × (−2) × (−3) = 12 (negative × negative is positive).½ mark
- a2 + 2ab + b2 = 4 + 12 + 9 = 25. Both sides are 25, so the equation works again.½ mark
(ii) Now suppose a and b are rational numbers, say a = −23 and b = 34.
- Left side: a + b = −23 + 34 = −812 + 912 = 112, so (a + b)2 = 1144.½ mark
- Right side: a2 = (−23)2 = 49; 2ab = 2 × (−23) × 34 = −1212 = −1; b2 = 916.½ mark
- a2 + 2ab + b2 = 49 − 1 + 916. Use the common denominator 144: 64 − 144 + 81144 = 145 − 144144 = 1144.½ mark
- Both sides are 1144, so the equation works for these fractions too.
Why it works for all numbers But we are still not sure if it is true for all numbers. To verify this, let us investigate further using the distributive property of numbers:
- (a + b)2 = (a + b)(a + b) = a(a + b) + b(a + b) = a2 + ab + ba + b2 = a2 + 2ab + b2.
- These steps use only rules that hold for every number, so the equation is true for all values of a and b. That is why it is called an identity.
Check: Try a = 5, b = −7: (5 − 7)2 = 4 and 25 + 2(5)(−7) + 49 = 25 − 70 + 49 = 4 ✓.
Answer to write in the exam
(i)
(a + b)2 = (−2 − 3)2 = (−5)2 = 25
a2 + 2ab + b2 = (−2)2 + 2(−2)(−3) + (−3)2 = 4 + 12 + 9 = 25
∴ (a + b)2 = a2 + 2ab + b2 = 25
(ii)
(a + b)2 = (−23 + 34)2 = (112)2 = 1144
a2 + 2ab + b2 = 49 + 2(−23)(34) + 916 = 49 − 1 + 916 = 64 − 144 + 81144 = 1144
∴ (a + b)2 = a2 + 2ab + b2 = 1144
Why it works for all numbers
(a + b)2 = (a + b)(a + b)
= a(a + b) + b(a + b) [distributive property]
= a2 + ab + ba + b2
= a2 + 2ab + b2
∴ (a + b)2 = a2 + 2ab + b2 for all a, b; it is an identity
Common mistakes that cost marks
- Writing (−2)2 = −4. A negative number squared is positive: (−2)2 = (−2) × (−2) = 4.
- Getting the sign of 2ab wrong. In (i) both numbers are negative, so 2ab = +12; in (ii) one is negative, so 2ab = −1.
- Adding fractions without a common denominator, e.g. −23 + 34 = 17. Use 12 as the common denominator: 112.
How this can come in the exam
If a = −4 and b = −1, the value of a2 + 2ab + b2 is
- 9
- 17
- 25
- −25
Show answer
(C) 25
It equals (a + b)2 = (−5)2 = 25. (16 + 8 + 1 = 25.)
Assertion (A): (a + b)2 = a2 + 2ab + b2 is true even when a and b are negative.
Reason (R): (a + b)2 = a2 + 2ab + b2 can be proved using the distributive property, which holds for all numbers.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
The distributive-property proof works for every number, positive, negative or fraction, so R explains A.
Try one yourself
Check that (a + b)2 = a2 + 2ab + b2 for a = 12 and b = −13.
Show answer
Left: (12 − 13)2 = (16)2 = 136. Right: 14 − 13 + 19 = 9 − 12 + 436 = 136. Both sides are 136 ✓.
More questions like this
- What is the difference between an equation and an identity?
- By now you must have observed that (a + b)2 ≠ a2 + b2. But can you find out which one of them will be greater? For example, if a = 10 and b = 2, what are the values of (a + b)2 and a2 + b2?
- 1. What can you say about a and b if (a + b)2 < a2 + b2?
2. What can you say about a and b if (a + b)2 > a2 + b2?
3. When will (a + b)2 be equal to a2 + b2?
Did you observe that (a + b)2 and a2 + b2 are both positive? What term will decide which is larger? Use the expansion of (a + b)2 to decide. - Let us try to expand (5x + 2y)2.
- To calculate 432, we can write it as (40 + 3)2.