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.
- 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?
Step-by-step solution
Idea: Expand: (a + b)2 = a2 + 2ab + b2. So (a + b)2 is just a2 + b2 with 2ab added. If 2ab is negative, the total shrinks; if positive, it grows; if zero, nothing changes. And the sign of ab depends only on the signs of a and b.
1. What can you say about a and b if (a + b)2 < a2 + b2?
- (a + b)2 < a2 + b2 means a2 + 2ab + b2 < a2 + b2, i.e. 2ab < 0, i.e. ab < 0.½ mark
- A product is negative only when one number is positive and the other is negative. Example: a = 5, b = −2: (3)2 = 9 < 25 + 4 = 29.½ mark
2. What can you say about a and b if (a + b)2 > a2 + b2?
- This means 2ab > 0, i.e. ab > 0.½ mark
- A product is positive when both numbers are positive or both are negative (and neither is 0). Example: a = −3, b = −4: (−7)2 = 49 > 9 + 16 = 25.½ mark
3. When will (a + b)2 be equal to a2 + b2?
- This means 2ab = 0, i.e. ab = 0.½ mark
- A product is 0 only when at least one of the numbers is 0. Example: a = 0, b = 6: (6)2 = 36 = 0 + 36.½ mark
Check: a = 10, b = 2 (same sign): 144 > 104 ✓. a = 10, b = −2 (opposite signs): 64 < 104 ✓. a = 10, b = 0: 100 = 100 ✓.
Answer to write in the exam
1.
(a + b)2 < a2 + b2
a2 + 2ab + b2 < a2 + b2
2ab < 0 ⇒ ab < 0
∴ a and b have opposite signs (one positive, one negative)
2.
(a + b)2 > a2 + b2
a2 + 2ab + b2 > a2 + b2
2ab > 0 ⇒ ab > 0
∴ a and b have the same sign, and neither is zero
3.
(a + b)2 = a2 + b2
a2 + 2ab + b2 = a2 + b2
2ab = 0 ⇒ ab = 0
∴ a = 0 or b = 0
Common mistakes that cost marks
- Saying “(a + b)2 is always bigger because squares are positive”. Both are positive, but 2ab can be negative.
- In part 2, saying “both positive” only. Both negative also works, because negative × negative is positive.
- In part 3, saying “a = b“. Equality needs ab = 0, i.e. one of them is zero, not that they are equal.
How this can come in the exam
For which pair is (a + b)2 < a2 + b2?
- a = 3, b = 4
- a = −3, b = −4
- a = 3, b = −4
- a = 0, b = 4
Show answer
(C) a = 3, b = −4
Only (C) has opposite signs, so 2ab = −24 < 0. Check: (−1)2 = 1 < 25.
Assertion (A): If a = −6 and b = −1, then (a + b)2 > a2 + b2.
Reason (R): (a + b)2 − (a2 + b2) = 2ab, which is positive when a and b are both negative.
- 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.
(−7)2 = 49 > 36 + 1 = 37, and the reason is that 2ab = 12 > 0.
Try one yourself
Without calculating the squares, decide which is larger: (8 + (−5))2 or 82 + (−5)2. By how much?
Show answer
2ab = 2 × 8 × (−5) = −80 < 0, so 82 + (−5)2 is larger, by 80. (Check: 32 = 9 and 64 + 25 = 89; 89 − 9 = 80.)
More questions like this
- Let us try to expand (5x + 2y)2.
- To calculate 432, we can write it as (40 + 3)2.
- Using the identity (a + b)2 = a2 + 2ab + b2, expand the following:
- Using the same identity, find the values of the following:
- The identity (a + b)2 = a2 + 2ab + b2 can also be used to find factors of some algebraic expressions. Consider the algebraic expression x2 + 4x + 4.