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If a + b + c = 0, then the quadratic equation 3ax2 + 2bx + c = 0 h as
- a)at least one root in [0, 1]
- b)one root in [2, 3] and the other in [ –2, –1]
- c)imaginary roots
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
If a + b + c = 0, then the quadratic equation 3ax2 + 2bx + c = 0 h asa...
Consider the function f (x) = ax3 + bx2 + cx on [0, 1] then being a polynomial. It is continuous on [0, 1], differentiable on (0, 1) and f (0) = f (1) = 0 [as given a + b + c = 0]
∴ By Rolle's theorem
such that 2
f '(x) = 0 ⇒ 3ax2 + 2bx +c = 0
Thus equation 3ax2 + 2bx + c = 0 has at least one root in [0, 1].
∴ By Rolle's theorem
such that 2f '(x) = 0 ⇒ 3ax2 + 2bx +c = 0
Thus equation 3ax2 + 2bx + c = 0 has at least one root in [0, 1].
Most Upvoted Answer
If a + b + c = 0, then the quadratic equation 3ax2 + 2bx + c = 0 h asa...
The correct options are missing in the question. Please provide the complete options so that the answer can be determined.
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If a + b + c = 0, then the quadratic equation 3ax2 + 2bx + c = 0 h asa...
The quadratic equation 3ax^2 + 2bx + c = 0 has at least one root in the interval [0, 1] if and only if the discriminant of the equation is greater than or equal to 0 and the vertex of the parabola represented by the equation is within the interval [0, 1].
The discriminant of the quadratic equation is given by b^2 - 4ac. Since a * b * c = 0, either a, b, or c must be equal to 0. If a = 0, then the quadratic equation becomes 2bx + c = 0, which is a linear equation with a single root. If b = 0, then the quadratic equation becomes 3ax^2 + c = 0, which is also a linear equation with a single root. If c = 0, then the quadratic equation becomes 3ax^2 + 2bx = 0, which can be factored as x(3ax + 2b) = 0, giving two roots at x = 0 and x = -2b/(3a).
Therefore, if a * b * c = 0, the quadratic equation 3ax^2 + 2bx + c = 0 will have at least one root in the interval [0, 1] if and only if either a = 0, b = 0, or c = 0.
The discriminant of the quadratic equation is given by b^2 - 4ac. Since a * b * c = 0, either a, b, or c must be equal to 0. If a = 0, then the quadratic equation becomes 2bx + c = 0, which is a linear equation with a single root. If b = 0, then the quadratic equation becomes 3ax^2 + c = 0, which is also a linear equation with a single root. If c = 0, then the quadratic equation becomes 3ax^2 + 2bx = 0, which can be factored as x(3ax + 2b) = 0, giving two roots at x = 0 and x = -2b/(3a).
Therefore, if a * b * c = 0, the quadratic equation 3ax^2 + 2bx + c = 0 will have at least one root in the interval [0, 1] if and only if either a = 0, b = 0, or c = 0.
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If a + b + c = 0, then the quadratic equation 3ax2 + 2bx + c = 0 h asa)at least one root in [0, 1]b)one root in [2, 3] and the other in [ –2, –1]c)imaginary rootsd)none of theseCorrect answer is option 'A'. Can you explain this answer?
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