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Let a > 0, b > 0 and c > 0. Then both the roots of the equation ax2 + bx + c = 0
- a)Are real and negative
- b)Have negative real parts
- c)Are rational numbers
- d)Have positive real parts
Correct answer is option 'B'. Can you explain this answer?
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Let a > 0, b > 0 and c > 0. Then both the roots of the equati...
Understanding the Roots of the Quadratic Equation
The quadratic equation given is ax² + bx + c = 0, where a, b, and c are all positive constants. To analyze the nature of the roots, we can use the properties of the quadratic formula and the discriminant.
Discriminant Analysis
- The discriminant (D) of the equation is calculated as D = b² - 4ac.
- Since a, b, and c are positive, the discriminant can potentially be positive, zero, or negative.
Roots of the Equation
- The roots of the quadratic equation are given by the formula: x = [ -b ± √D ] / (2a).
- If D > 0, the roots are real and distinct.
- If D = 0, the roots are real and equal.
- If D < 0,="" the="" roots="" are="" complex="" conjugates.="" />Real Parts of the Roots
- Regardless of whether the roots are real or complex, the term -b will always be negative since b > 0.
- Therefore, both roots will have a negative real part:
- For real roots: Since -b is negative, the entire expression for the roots will yield negative values.
- For complex roots, the real part is -b, which is also negative.
Conclusion
- Thus, both roots of the quadratic equation ax² + bx + c = 0, given that a > 0, b > 0, and c > 0, will have negative real parts. Hence, the correct option is B.
This analysis confirms the nature of the roots under the given conditions, making option B the valid choice.
The quadratic equation given is ax² + bx + c = 0, where a, b, and c are all positive constants. To analyze the nature of the roots, we can use the properties of the quadratic formula and the discriminant.
Discriminant Analysis
- The discriminant (D) of the equation is calculated as D = b² - 4ac.
- Since a, b, and c are positive, the discriminant can potentially be positive, zero, or negative.
Roots of the Equation
- The roots of the quadratic equation are given by the formula: x = [ -b ± √D ] / (2a).
- If D > 0, the roots are real and distinct.
- If D = 0, the roots are real and equal.
- If D < 0,="" the="" roots="" are="" complex="" conjugates.="" />Real Parts of the Roots
- Regardless of whether the roots are real or complex, the term -b will always be negative since b > 0.
- Therefore, both roots will have a negative real part:
- For real roots: Since -b is negative, the entire expression for the roots will yield negative values.
- For complex roots, the real part is -b, which is also negative.
Conclusion
- Thus, both roots of the quadratic equation ax² + bx + c = 0, given that a > 0, b > 0, and c > 0, will have negative real parts. Hence, the correct option is B.
This analysis confirms the nature of the roots under the given conditions, making option B the valid choice.
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Let a > 0, b > 0 and c > 0. Then both the roots of the equation ax2+ bx + c = 0a)Are real and negativeb)Have negative real partsc)Are rational numbersd)Have positive real partsCorrect answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let a > 0, b > 0 and c > 0. Then both the roots of the equation ax2+ bx + c = 0a)Are real and negativeb)Have negative real partsc)Are rational numbersd)Have positive real partsCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let a > 0, b > 0 and c > 0. Then both the roots of the equation ax2+ bx + c = 0a)Are real and negativeb)Have negative real partsc)Are rational numbersd)Have positive real partsCorrect answer is option 'B'. Can you explain this answer?.
Let a > 0, b > 0 and c > 0. Then both the roots of the equation ax2+ bx + c = 0a)Are real and negativeb)Have negative real partsc)Are rational numbersd)Have positive real partsCorrect answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let a > 0, b > 0 and c > 0. Then both the roots of the equation ax2+ bx + c = 0a)Are real and negativeb)Have negative real partsc)Are rational numbersd)Have positive real partsCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let a > 0, b > 0 and c > 0. Then both the roots of the equation ax2+ bx + c = 0a)Are real and negativeb)Have negative real partsc)Are rational numbersd)Have positive real partsCorrect answer is option 'B'. Can you explain this answer?.
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