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Consider the quadratic equation (a + c − b)x2 + 2cx + (b + c − a) = 0, where a, b and c are distinct rational numbers and a + c − b ≠ 0. Then,
  • a)
    both roots are rational
  • b)
    both roots are integers
  • c)
    both roots are real
  • d)
    one root is positive and one root is negative
Correct answer is option 'A,C'. Can you explain this answer?
Most Upvoted Answer
Consider the quadratic equation (a + c − b)x2 + 2cx + (b + c − a) = 0...
For the given quadratic equation (a + c − b)x2 + 2cx + (b + c − a) = 0, discriminant = 4c2 − 4(a + c − b)(b + c − a) = 4(a − b)2, which is greater than equal to zero and is a perfect square of a rational number.
Hence, the roots are real.
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Community Answer
Consider the quadratic equation (a + c − b)x2 + 2cx + (b + c − a) = 0...
Explanation:

Rational Roots:
- The given quadratic equation has coefficients that are rational numbers, and since the roots of a quadratic equation with rational coefficients are either rational or irrational conjugates, it follows that both roots are rational.

Real Roots:
- For a quadratic equation to have real roots, the discriminant (b^2 - 4ac) must be greater than or equal to 0. In this case, the discriminant is 4c^2 - 4(a + c - b)(b + c - a) = 4c^2 - 4(ab + ac - b^2 - ab - ac + a^2 + bc - ac) = 4c^2 - 4(a^2 - b^2 + bc - ac) = 4c^2 - 4(a - b)(a + b - c).
- Since a, b, and c are distinct rational numbers, the discriminant will always be greater than 0, ensuring that both roots are real.
Therefore, both roots of the given quadratic equation are rational and real.
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Consider the quadratic equation (a + c − b)x2 + 2cx + (b + c − a) = 0, where a, b and c are distinct rational numbers and a + c − b ≠ 0. Then,a)both roots are rationalb)both roots are integersc)both roots are reald)one root is positive and one root is negativeCorrect answer is option 'A,C'. Can you explain this answer?
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Consider the quadratic equation (a + c − b)x2 + 2cx + (b + c − a) = 0, where a, b and c are distinct rational numbers and a + c − b ≠ 0. Then,a)both roots are rationalb)both roots are integersc)both roots are reald)one root is positive and one root is negativeCorrect answer is option 'A,C'. 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 Consider the quadratic equation (a + c − b)x2 + 2cx + (b + c − a) = 0, where a, b and c are distinct rational numbers and a + c − b ≠ 0. Then,a)both roots are rationalb)both roots are integersc)both roots are reald)one root is positive and one root is negativeCorrect answer is option 'A,C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the quadratic equation (a + c − b)x2 + 2cx + (b + c − a) = 0, where a, b and c are distinct rational numbers and a + c − b ≠ 0. Then,a)both roots are rationalb)both roots are integersc)both roots are reald)one root is positive and one root is negativeCorrect answer is option 'A,C'. Can you explain this answer?.
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