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A quadratic equation f(x) = 0 has one positive and one negative root. If the graph of f(x) is shifted left along the x-axis to form f'(x) in such a way that the sum of roots of the quadratic equation f'(x) = 0 is zero, the distance between the minima of f(x) and f'(x) is 5 units. Which of the following is a possible sum of the roots of the equation f(x) = 0?
  • a)
    10
  • b)
    5
  • c)
    2.5
  • d)
    -2.5
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
A quadratic equation f(x) = 0 has one positive and one negative root. ...
Let the roots of f'(x) = -a,a
The distance between the minima of f(x) and f'(x) = 5. This implies that f(x) has been shifted 5 units towards the left to get f'(x).
Hence, the roots of f(x) = -a+5 and a+5.
Hence, sum of roots of f(x) = -a+5+a+5 = 10 units.
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Most Upvoted Answer
A quadratic equation f(x) = 0 has one positive and one negative root. ...
Given information:
- A quadratic equation f(x) = 0 has one positive and one negative root.
- The graph of f(x) is shifted left along the x-axis to form f(x) in such a way that the sum of roots of the quadratic equation f(x) = 0 is zero.
- The distance between the minima of f(x) and f(x) is 5 units.

To find: The possible sum of the roots of the equation f(x) = 0.

Analysis:
- Shifting the graph of f(x) to the left along the x-axis means that the roots of the equation f(x) = 0 will also shift to the left.
- The sum of the roots of a quadratic equation is equal to the negation of the coefficient of x in the linear term divided by the coefficient of x^2 in the quadratic term.
- The distance between the minima of f(x) and f(x) is 5 units. This means that the vertex of the parabola representing f(x) will shift 5 units to the left.

Solution:
Let's assume that the original quadratic equation f(x) = ax^2 + bx + c has roots α and β.
The sum of the roots, α + β, is given by -b/a.
Since the graph is shifted to the left, the new quadratic equation f(x) = a(x + 5)^2 + b(x + 5) + c will have roots α - 5 and β - 5.
The sum of the new roots, (α - 5) + (β - 5), is equal to α + β - 10.

To satisfy the condition that the sum of the roots is zero, we have:
α + β - 10 = 0
α + β = 10

Therefore, the possible sum of the roots of the equation f(x) = 0 is 10, which corresponds to option A.
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A quadratic equation f(x) = 0 has one positive and one negative root. If the graph of f(x) is shifted left along the x-axis to form f(x) in such a way that the sum of roots of the quadratic equation f(x) = 0 is zero, the distance between the minima of f(x) and f(x) is 5 units. Which of the following is a possible sum of the roots of the equation f(x) = 0?a)10b)5c)2.5d)-2.5Correct answer is option 'A'. Can you explain this answer?
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A quadratic equation f(x) = 0 has one positive and one negative root. If the graph of f(x) is shifted left along the x-axis to form f(x) in such a way that the sum of roots of the quadratic equation f(x) = 0 is zero, the distance between the minima of f(x) and f(x) is 5 units. Which of the following is a possible sum of the roots of the equation f(x) = 0?a)10b)5c)2.5d)-2.5Correct answer is option 'A'. 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 A quadratic equation f(x) = 0 has one positive and one negative root. If the graph of f(x) is shifted left along the x-axis to form f(x) in such a way that the sum of roots of the quadratic equation f(x) = 0 is zero, the distance between the minima of f(x) and f(x) is 5 units. Which of the following is a possible sum of the roots of the equation f(x) = 0?a)10b)5c)2.5d)-2.5Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A quadratic equation f(x) = 0 has one positive and one negative root. If the graph of f(x) is shifted left along the x-axis to form f(x) in such a way that the sum of roots of the quadratic equation f(x) = 0 is zero, the distance between the minima of f(x) and f(x) is 5 units. Which of the following is a possible sum of the roots of the equation f(x) = 0?a)10b)5c)2.5d)-2.5Correct answer is option 'A'. Can you explain this answer?.
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