What is the sum of the roots of all the quadr...
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)
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).
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.
Related Test
 Test: Algebra - 130 Ques | 50 Mins
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).

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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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