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If one root of ax^2 + bx + c = 0 is n times the other root then prove that ac (1 +n)^2 - b^2n = 0?
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If one root of ax^2 + bx + c = 0 is n times the other root then prove ...
To prove that ac(1 + n)^2 - b^2n = 0 when one root of the quadratic equation ax^2 + bx + c = 0 is n times the other root, we can follow these steps:
Understanding the Roots
- Let the roots of the equation be α and nα, where α is one root and nα is the other root.
Using Vieta's Formulas
- According to Vieta’s formulas:
- The sum of the roots (α + nα) = -b/a
- The product of the roots (α * nα) = c/a
Expressing Sum and Product
- From the sum of the roots:
- α(1 + n) = -b/a
- Therefore, α = -b/a(1 + n)
- From the product of the roots:
- nα^2 = c/a
Substituting and Rearranging
- Substitute α into the product equation:
- n(-b/a(1 + n))^2 = c/a
- Simplifying gives: n(b^2/(a^2(1 + n)^2)) = c/a
Cross-Multiplying
- Cross-multiplying results in:
- b^2n = ac(1 + n)^2
Final Conclusion
- Rearranging leads to:
- ac(1 + n)^2 - b^2n = 0
Thus, we have successfully proven that when one root is n times the other, the equation ac(1 + n)^2 - b^2n = 0 holds true.
Understanding the Roots
- Let the roots of the equation be α and nα, where α is one root and nα is the other root.
Using Vieta's Formulas
- According to Vieta’s formulas:
- The sum of the roots (α + nα) = -b/a
- The product of the roots (α * nα) = c/a
Expressing Sum and Product
- From the sum of the roots:
- α(1 + n) = -b/a
- Therefore, α = -b/a(1 + n)
- From the product of the roots:
- nα^2 = c/a
Substituting and Rearranging
- Substitute α into the product equation:
- n(-b/a(1 + n))^2 = c/a
- Simplifying gives: n(b^2/(a^2(1 + n)^2)) = c/a
Cross-Multiplying
- Cross-multiplying results in:
- b^2n = ac(1 + n)^2
Final Conclusion
- Rearranging leads to:
- ac(1 + n)^2 - b^2n = 0
Thus, we have successfully proven that when one root is n times the other, the equation ac(1 + n)^2 - b^2n = 0 holds true.
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If one root of ax^2 + bx + c = 0 is n times the other root then prove that ac (1 +n)^2 - b^2n = 0? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If one root of ax^2 + bx + c = 0 is n times the other root then prove that ac (1 +n)^2 - b^2n = 0? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If one root of ax^2 + bx + c = 0 is n times the other root then prove that ac (1 +n)^2 - b^2n = 0?.
If one root of ax^2 + bx + c = 0 is n times the other root then prove that ac (1 +n)^2 - b^2n = 0? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If one root of ax^2 + bx + c = 0 is n times the other root then prove that ac (1 +n)^2 - b^2n = 0? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If one root of ax^2 + bx + c = 0 is n times the other root then prove that ac (1 +n)^2 - b^2n = 0?.
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