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What is the sum of the roots of all the quadratic equations that can be formed such that both the roots of the quadratic equation are common with the roots of equation (x – a) (x – b) (x – c) = 0?
(2014)
- a)3(a + b + c)
- b)2(a + b + c)
- c)(a + b + c)
- d)4(a + b + c)
Correct answer is option 'D'. Can you explain this answer?
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What is the sum of the roots of all the quadratic equations that can b...
The equations formed by the roots of the equation
(x – a) (x – b)(x – c) can be as follows:
(i) (x – a)(x – b) ⇒ Roots are a, b
(ii) (x – b)(x – c) ⇒ Roots are b, c
(iii) (x – c)(x – a) ⇒ Roots are c, a
(iv) (x – a)2 ⇒ Roots are a, a
(v) (x – b)2 ⇒ Roots are b, b
(vi) (x – c)2 ⇒ Roots are c, c
Adding all these roots, we get 4(a + b + c).
(x – a) (x – b)(x – c) can be as follows:
(i) (x – a)(x – b) ⇒ Roots are a, b
(ii) (x – b)(x – c) ⇒ Roots are b, c
(iii) (x – c)(x – a) ⇒ Roots are c, a
(iv) (x – a)2 ⇒ Roots are a, a
(v) (x – b)2 ⇒ Roots are b, b
(vi) (x – c)2 ⇒ Roots are c, c
Adding all these roots, we get 4(a + b + c).
Most Upvoted Answer
What is the sum of the roots of all the quadratic equations that can b...
The quadratic equation (x - a)(x - b) = 0, where a and b are the roots, can be expanded as x^2 - (a + b)x + ab = 0.
For the quadratic equation to have roots in common with (x - a)(x - b) = 0, the discriminant of (x - a)(x - b) = 0, which is (a + b)^2 - 4ab, must be zero.
Setting the discriminant to zero, we have (a + b)^2 - 4ab = 0.
Expanding this equation, we get a^2 + 2ab + b^2 - 4ab = 0.
Simplifying, we have a^2 - 2ab + b^2 = 0.
This equation can be factored as (a - b)^2 = 0.
Taking the square root of both sides, we have a - b = 0.
Therefore, the sum of the roots of all the quadratic equations that can be formed such that both the roots of the quadratic equation are common with the roots of (x - a)(x - b) = 0 is a + b.
So, the sum of the roots is a + b.
For the quadratic equation to have roots in common with (x - a)(x - b) = 0, the discriminant of (x - a)(x - b) = 0, which is (a + b)^2 - 4ab, must be zero.
Setting the discriminant to zero, we have (a + b)^2 - 4ab = 0.
Expanding this equation, we get a^2 + 2ab + b^2 - 4ab = 0.
Simplifying, we have a^2 - 2ab + b^2 = 0.
This equation can be factored as (a - b)^2 = 0.
Taking the square root of both sides, we have a - b = 0.
Therefore, the sum of the roots of all the quadratic equations that can be formed such that both the roots of the quadratic equation are common with the roots of (x - a)(x - b) = 0 is a + b.
So, the sum of the roots is a + b.
Community Answer
What is the sum of the roots of all the quadratic equations that can b...
The equations formed by the roots of the equation
(x – a) (x – b)(x – c) can be as follows:
(i) (x – a)(x – b) ⇒ Roots are a, b
(ii) (x – b)(x – c) ⇒ Roots are b, c
(iii) (x – c)(x – a) ⇒ Roots are c, a
(iv) (x – a)2 ⇒ Roots are a, a
(v) (x – b)2 ⇒ Roots are b, b
(vi) (x – c)2 ⇒ Roots are c, c
Adding all these roots, we get 4(a + b + c).
(x – a) (x – b)(x – c) can be as follows:
(i) (x – a)(x – b) ⇒ Roots are a, b
(ii) (x – b)(x – c) ⇒ Roots are b, c
(iii) (x – c)(x – a) ⇒ Roots are c, a
(iv) (x – a)2 ⇒ Roots are a, a
(v) (x – b)2 ⇒ Roots are b, b
(vi) (x – c)2 ⇒ Roots are c, c
Adding all these roots, we get 4(a + b + c).
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What is the sum of the roots of all the quadratic equations that can be formed such that both the roots of the quadratic equation are common with the roots of equation (x – a) (x – b) (x – c) = 0?(2014)a)3(a + b + c)b)2(a + b + c)c)(a + b + c)d)4(a + b + c)Correct answer is option 'D'. Can you explain this answer?
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What is the sum of the roots of all the quadratic equations that can be formed such that both the roots of the quadratic equation are common with the roots of equation (x – a) (x – b) (x – c) = 0?(2014)a)3(a + b + c)b)2(a + b + c)c)(a + b + c)d)4(a + b + c)Correct answer is option 'D'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about What is the sum of the roots of all the quadratic equations that can be formed such that both the roots of the quadratic equation are common with the roots of equation (x – a) (x – b) (x – c) = 0?(2014)a)3(a + b + c)b)2(a + b + c)c)(a + b + c)d)4(a + b + c)Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the sum of the roots of all the quadratic equations that can be formed such that both the roots of the quadratic equation are common with the roots of equation (x – a) (x – b) (x – c) = 0?(2014)a)3(a + b + c)b)2(a + b + c)c)(a + b + c)d)4(a + b + c)Correct answer is option 'D'. Can you explain this answer?.
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