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If the roots of the equation bx2 + cx + a = 0 be imaginary, then for all real values of x, the expression 3b2x2 + 6bcx + 2c2 is : [2009]
- a)less than 4ab
- b)greater than – 4ab
- c)1ess than – 4ab
- d)greater than 4ab
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
If the roots of the equation bx2 + cx + a = 0 be imaginary, then for a...
Given that roots of the equation bx2 + cx + a = 0 are imaginary
∴ c2 – 4ab < 0 ....(i)
∴ c2 – 4ab < 0 ....(i)
Let y = 3b2x2 + 6 bc x + 2c2
⇒ 3b2x2 + 6 bc x + 2c2 – y = 0
As x is real, D ≥ 0
⇒ 36 b2c2 – 12 b2 (2c2 – y ) ≥ 0
⇒ 12 b2 (3 c2 – 2 c2+ y ) ≥ 0
⇒ c2 + y ≥ 0
⇒ y ≥ – c2
But from eqn.
(i), c2 < 4ab or – c2 > – 4ab
∴ we get y ≥ – c2 > – 4ab
⇒ y > – 4 ab
⇒ 3b2x2 + 6 bc x + 2c2 – y = 0
As x is real, D ≥ 0
⇒ 36 b2c2 – 12 b2 (2c2 – y ) ≥ 0
⇒ 12 b2 (3 c2 – 2 c2+ y ) ≥ 0
⇒ c2 + y ≥ 0
⇒ y ≥ – c2
But from eqn.
(i), c2 < 4ab or – c2 > – 4ab
∴ we get y ≥ – c2 > – 4ab
⇒ y > – 4 ab
Most Upvoted Answer
If the roots of the equation bx2 + cx + a = 0 be imaginary, then for a...
To determine the relationship between the expression 3b^2x^2 + 6bcx + 2c^2 and 4ab, we can start by factoring out a common factor of 3 from the expression:
3b^2x^2 + 6bcx + 2c^2 = 3(b^2x^2 + 2bcx + 2/3 c^2)
Next, we can complete the square for the quadratic expression in the parentheses:
b^2x^2 + 2bcx + (bc)^2 = (bx + bc)^2
Substituting this back into the expression, we have:
3(bx + bc)^2 + 2c^2
Since the roots of the equation bx^2 + cx + a = 0 are imaginary, the discriminant b^2 - 4ac must be negative:
c^2 - 4ab < />
Rearranging this inequality, we have:
4ab > c^2
Substituting this inequality into the expression, we get:
3(bx + bc)^2 + 2c^2 > 3(bx + bc)^2 + 4ab
Since 3(bx + bc)^2 + 4ab is greater than the expression 3b^2x^2 + 6bcx + 2c^2, we can conclude that:
3b^2x^2 + 6bcx + 2c^2 < />
Therefore, the correct answer is:
a) less than 4ab.
3b^2x^2 + 6bcx + 2c^2 = 3(b^2x^2 + 2bcx + 2/3 c^2)
Next, we can complete the square for the quadratic expression in the parentheses:
b^2x^2 + 2bcx + (bc)^2 = (bx + bc)^2
Substituting this back into the expression, we have:
3(bx + bc)^2 + 2c^2
Since the roots of the equation bx^2 + cx + a = 0 are imaginary, the discriminant b^2 - 4ac must be negative:
c^2 - 4ab < />
Rearranging this inequality, we have:
4ab > c^2
Substituting this inequality into the expression, we get:
3(bx + bc)^2 + 2c^2 > 3(bx + bc)^2 + 4ab
Since 3(bx + bc)^2 + 4ab is greater than the expression 3b^2x^2 + 6bcx + 2c^2, we can conclude that:
3b^2x^2 + 6bcx + 2c^2 < />
Therefore, the correct answer is:
a) less than 4ab.
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If the roots of the equation bx2 + cx + a = 0 be imaginary, then for all real values of x, the expression 3b2x2 + 6bcx + 2c2 is : [2009]a)less than 4abb)greater than – 4abc)1ess than – 4abd)greater than 4abCorrect answer is option 'B'. Can you explain this answer?
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If the roots of the equation bx2 + cx + a = 0 be imaginary, then for all real values of x, the expression 3b2x2 + 6bcx + 2c2 is : [2009]a)less than 4abb)greater than – 4abc)1ess than – 4abd)greater than 4abCorrect answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the roots of the equation bx2 + cx + a = 0 be imaginary, then for all real values of x, the expression 3b2x2 + 6bcx + 2c2 is : [2009]a)less than 4abb)greater than – 4abc)1ess than – 4abd)greater than 4abCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the roots of the equation bx2 + cx + a = 0 be imaginary, then for all real values of x, the expression 3b2x2 + 6bcx + 2c2 is : [2009]a)less than 4abb)greater than – 4abc)1ess than – 4abd)greater than 4abCorrect answer is option 'B'. Can you explain this answer?.
If the roots of the equation bx2 + cx + a = 0 be imaginary, then for all real values of x, the expression 3b2x2 + 6bcx + 2c2 is : [2009]a)less than 4abb)greater than – 4abc)1ess than – 4abd)greater than 4abCorrect answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the roots of the equation bx2 + cx + a = 0 be imaginary, then for all real values of x, the expression 3b2x2 + 6bcx + 2c2 is : [2009]a)less than 4abb)greater than – 4abc)1ess than – 4abd)greater than 4abCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the roots of the equation bx2 + cx + a = 0 be imaginary, then for all real values of x, the expression 3b2x2 + 6bcx + 2c2 is : [2009]a)less than 4abb)greater than – 4abc)1ess than – 4abd)greater than 4abCorrect answer is option 'B'. Can you explain this answer?.
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