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If (a + b)2 – 2(a + b) = 80 and ab = 16, then what can be the value of 3a – 19b?
  • a)
    -16
  • b)
    -14
  • c)
    -18
  • d)
    -20
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
If (a + b)2– 2(a + b) = 80 and ab = 16, then what can be the val...
(a + b)2 – 2(a + b) = 80
⇒ (a + b)2 – 2(a + b) + 1 = 81
⇒ (a + b – 1)2 = 81
⇒ a + b – 1 = 9
⇒ a + b = 10
given ab = 16
⇒ a = 8 and b = 2
⇒ 3a – 19b = 24 – 38 = -14
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If (a + b)2– 2(a + b) = 80 and ab = 16, then what can be the val...
Given Equations
We start with the equations:
- (a + b)² - 2(a + b) = 80
- ab = 16

Step 1: Simplify the First Equation
We can rewrite the first equation:
- Let x = a + b.
- The equation becomes x² - 2x = 80.
Rearranging gives us:
- x² - 2x - 80 = 0.

Step 2: Solve the Quadratic Equation
To find x, we can use the quadratic formula:
- x = [-b ± √(b² - 4ac)] / 2a
- Here, a = 1, b = -2, c = -80.
Calculating the discriminant:
- Discriminant = (-2)² - 4(1)(-80) = 4 + 320 = 324.
Thus:
- x = [2 ± √324] / 2
- x = [2 ± 18] / 2.
This gives us:
- x = 10 or x = -8.

Step 3: Determine a and b
We have two cases for (a + b):
1. If a + b = 10, then:
- ab = 16.
- The roots are found using: t² - (a + b)t + ab = 0.
- t² - 10t + 16 = 0.
- Discriminant = 100 - 64 = 36.
- Roots are: t = 8 and t = 2, thus (a, b) = (8, 2) or (2, 8).
2. If a + b = -8, then:
- ab = 16.
- t² + 8t + 16 = 0.
- Discriminant = 64 - 64 = 0.
- Roots are: t = -4 (double root), thus (a, b) = (-4, -4).

Step 4: Calculate 3a - 19b
1. For (a, b) = (8, 2):
- 3a - 19b = 3(8) - 19(2) = 24 - 38 = -14.
2. For (a, b) = (-4, -4):
- 3(-4) - 19(-4) = -12 + 76 = 64 (not valid).

Final Answer
Thus, the valid value for 3a - 19b is:
- **-14**, which corresponds to option (B).
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If (a + b)2– 2(a + b) = 80 and ab = 16, then what can be the value of 3a – 19b?a)-16b)-14c)-18d)-20Correct answer is option 'B'. Can you explain this answer?
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