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If a, b, c are integers and b2 = 4(ac + 5d2), d ∈ N, then roots of the equation ax2 + bx + c = 0 are
- a)Rational & defferent
- b)Complex conjugate
- c)Irrational
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
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If a, b, c are integers and b2= 4(ac + 5d2), d ∈ N, then roots of...
Given Information
- We are given the equation \(b^2 = 4(ac + 5d^2)\) where \(a\), \(b\), \(c\) are integers and \(d \in \mathbb{N}\).
- We need to find the roots of the equation \(ax^2 + bx + c = 0\).
Explanation
- For the roots of the quadratic equation to be irrational, the discriminant (\(b^2 - 4ac\)) must be negative.
- From the given equation, we have \(b^2 = 4ac + 20d^2\).
- Rearranging terms, we get \(b^2 - 20d^2 = 4ac\).
- Substituting this into the discriminant, we have \(\Delta = b^2 - 4ac = 20d^2\).
- Since \(d^2\) is a square of a natural number, the value of \(20d^2\) will always be a multiple of 20.
- As a result, the discriminant will always be a multiple of 20, which means it will be non-negative.
- Therefore, the roots of the quadratic equation will be irrational.
Conclusion
Hence, the roots of the equation \(ax^2 + bx + c = 0\) will be irrational.
- We are given the equation \(b^2 = 4(ac + 5d^2)\) where \(a\), \(b\), \(c\) are integers and \(d \in \mathbb{N}\).
- We need to find the roots of the equation \(ax^2 + bx + c = 0\).
Explanation
- For the roots of the quadratic equation to be irrational, the discriminant (\(b^2 - 4ac\)) must be negative.
- From the given equation, we have \(b^2 = 4ac + 20d^2\).
- Rearranging terms, we get \(b^2 - 20d^2 = 4ac\).
- Substituting this into the discriminant, we have \(\Delta = b^2 - 4ac = 20d^2\).
- Since \(d^2\) is a square of a natural number, the value of \(20d^2\) will always be a multiple of 20.
- As a result, the discriminant will always be a multiple of 20, which means it will be non-negative.
- Therefore, the roots of the quadratic equation will be irrational.
Conclusion
Hence, the roots of the equation \(ax^2 + bx + c = 0\) will be irrational.
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If a, b, c are integers and b2= 4(ac + 5d2), d ∈ N, then roots of the equation ax2+ bx + c = 0 area)Rational & defferentb)Complex conjugatec)Irrationald)none of theseCorrect answer is option 'C'. Can you explain this answer?
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