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The quadratic equation x2 + bx + c = 0 has two roots 4a and 3a, where a is an integer. Which of the following is a possible value of b2 + c?
(2019)
- a)3721
- b)361
- c)549
- d)427
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
The quadratic equation x2 + bx + c = 0 has two roots 4a and 3a, where ...
Sum of roots = 4a + 3a = 7a = –b
∴ b = –7a
Product of roots = 4a × 3a = c
∴ c = 12a2
Now, b2 + c = (–7a)2 + 12a2 = 61a2
Comparing the options
Option (a) : 61a2 = 3721 ⇒ a2 = 61, clearly a is not an integer.
Option (b) : 61a2 = 361 ⇒ a2 = 361 / 61, clearly a is not an integer.
Option (c) : 61a2 = 549 ⇒ a2 = 9, we can have a = –3 or 3 (an integer)
Option (d) : 61a2 = 427 ⇒ a2 = 7, clearly a is not an integer
∴ b = –7a
Product of roots = 4a × 3a = c
∴ c = 12a2
Now, b2 + c = (–7a)2 + 12a2 = 61a2
Comparing the options
Option (a) : 61a2 = 3721 ⇒ a2 = 61, clearly a is not an integer.
Option (b) : 61a2 = 361 ⇒ a2 = 361 / 61, clearly a is not an integer.
Option (c) : 61a2 = 549 ⇒ a2 = 9, we can have a = –3 or 3 (an integer)
Option (d) : 61a2 = 427 ⇒ a2 = 7, clearly a is not an integer
Most Upvoted Answer
The quadratic equation x2 + bx + c = 0 has two roots 4a and 3a, where ...
Given quadratic equation: x^2 + bx + c = 0
The roots of the equation are 4a and 3a.
Product of roots = 4a * 3a = 12a^2
Sum of roots = 4a + 3a = 7a
Therefore, the quadratic equation can be written as:
x^2 - (sum of roots)x + (product of roots) = 0
x^2 - 7ax + 12a^2 = 0
Comparing this equation with the given equation: x^2 + bx + c = 0
We can conclude that b = -7a and c = 12a^2
To find the value of b^2 - c, we substitute the values of b and c:
b^2 - c = (-7a)^2 - 12a^2
= 49a^2 - 12a^2
= 37a^2
Since a is an integer, we can conclude that b^2 - c is a multiple of 37.
Now, let's check the given options:
a) 3721 = 37 * 101
b) 361 = 19 * 19
c) 549 = 3 * 183
d) 427 = 7 * 61
Among the given options, only option c) 549 is a multiple of 37.
Therefore, the correct answer is option c) 549.
The roots of the equation are 4a and 3a.
Product of roots = 4a * 3a = 12a^2
Sum of roots = 4a + 3a = 7a
Therefore, the quadratic equation can be written as:
x^2 - (sum of roots)x + (product of roots) = 0
x^2 - 7ax + 12a^2 = 0
Comparing this equation with the given equation: x^2 + bx + c = 0
We can conclude that b = -7a and c = 12a^2
To find the value of b^2 - c, we substitute the values of b and c:
b^2 - c = (-7a)^2 - 12a^2
= 49a^2 - 12a^2
= 37a^2
Since a is an integer, we can conclude that b^2 - c is a multiple of 37.
Now, let's check the given options:
a) 3721 = 37 * 101
b) 361 = 19 * 19
c) 549 = 3 * 183
d) 427 = 7 * 61
Among the given options, only option c) 549 is a multiple of 37.
Therefore, the correct answer is option c) 549.
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The quadratic equation x2 + bx + c = 0 has two roots 4a and 3a, where a is an integer. Which of the following is a possible value of b2 + c?(2019)a)3721b)361c)549d)427Correct answer is option 'C'. Can you explain this answer?
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