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If a,b,c are odd integers; prove that the equation ax²+bx+c=0 can not have rational roots?
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If a,b,c are odd integers; prove that the equation ax²+bx+c=0 can not ...
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If a,b,c are odd integers; prove that the equation ax²+bx+c=0 can not ...
Proof that the equation ax²+bx+c=0 cannot have rational roots when a, b, c are odd integers:
1. Assume the equation has rational roots:
- Let's assume that the equation ax² + bx + c = 0 has rational roots p/q, where p and q are coprime integers.
2. Substitute the roots into the equation:
- Substituting the rational roots into the equation, we get ap²/q² + bp/q + c = 0.
3. Clearing the denominator:
- Multiplying through by q² to clear the denominator, we get ap² + bpq + cq² = 0.
4. Looking at the equation modulo 2:
- Since a, b, and c are odd integers, ap² ≡ a (mod 2), bpq ≡ b (mod 2), and cq² ≡ c (mod 2).
- Therefore, the equation simplifies to a + b + c ≡ 0 (mod 2).
5. Contradiction:
- Since a, b, and c are odd integers, their sum a + b + c is odd.
- But 0 is even, leading to a contradiction.
6. Conclusion:
- The assumption that the equation has rational roots leads to a contradiction.
- Therefore, the equation ax² + bx + c = 0 cannot have rational roots when a, b, and c are odd integers.
1. Assume the equation has rational roots:
- Let's assume that the equation ax² + bx + c = 0 has rational roots p/q, where p and q are coprime integers.
2. Substitute the roots into the equation:
- Substituting the rational roots into the equation, we get ap²/q² + bp/q + c = 0.
3. Clearing the denominator:
- Multiplying through by q² to clear the denominator, we get ap² + bpq + cq² = 0.
4. Looking at the equation modulo 2:
- Since a, b, and c are odd integers, ap² ≡ a (mod 2), bpq ≡ b (mod 2), and cq² ≡ c (mod 2).
- Therefore, the equation simplifies to a + b + c ≡ 0 (mod 2).
5. Contradiction:
- Since a, b, and c are odd integers, their sum a + b + c is odd.
- But 0 is even, leading to a contradiction.
6. Conclusion:
- The assumption that the equation has rational roots leads to a contradiction.
- Therefore, the equation ax² + bx + c = 0 cannot have rational roots when a, b, and c are odd integers.
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If a,b,c are odd integers; prove that the equation ax²+bx+c=0 can not have rational roots?
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If a,b,c are odd integers; prove that the equation ax²+bx+c=0 can not have rational roots? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If a,b,c are odd integers; prove that the equation ax²+bx+c=0 can not have rational roots? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a,b,c are odd integers; prove that the equation ax²+bx+c=0 can not have rational roots?.
If a,b,c are odd integers; prove that the equation ax²+bx+c=0 can not have rational roots? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If a,b,c are odd integers; prove that the equation ax²+bx+c=0 can not have rational roots? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a,b,c are odd integers; prove that the equation ax²+bx+c=0 can not have rational roots?.
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