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If one of the root of a quadratic equation with rational coefficients is rational, then other root must be
- a)Imaginary
- b)Irrational
- c)Rational
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
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If one of the root of a quadratic equation with rational coefficients ...
Also, αβ = r/p, which is also rational. α + β = (a+√b) + (a-√b) = 2a, a rational number and, αβ = (a+√b)(a-√b) = a² - b, a rational number. So, the other root of a quadratic equation having the one root as (a+√b) is (a-√b), where a and b are rational numbers.
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If one of the root of a quadratic equation with rational coefficients ...
Roots occurs along with their conjugatese.g. irrat
ional root occurs with its conjugate and imaginary root occur wit
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If one of the root of a quadratic equation with rational coefficients ...
Explanation:
When a quadratic equation has rational coefficients, it can have roots that are either rational or irrational. Let's consider a quadratic equation:
ax^2 + bx + c = 0
If one of the roots is rational, say p/q, where p and q are integers with no common factors, then the other root must be of the form:
(-b + sqrt(b^2 - 4ac))/2a or (-b - sqrt(b^2 - 4ac))/2a
Let's assume that (-b + sqrt(b^2 - 4ac))/2a is the other root. We know that the sum of the roots is -b/a and the product of the roots is c/a. Therefore,
(-b + p/q) + (-b + sqrt(b^2 - 4ac))/2a = -b/a
Simplifying the above equation, we get:
sqrt(b^2 - 4ac) = (2ap - bq)/q
Since p/q is rational and a, b, c are rational coefficients, (2ap - bq)/q must be a rational number. Therefore, sqrt(b^2 - 4ac) must be a rational number as well. This implies that the other root must be of the form:
(-b - sqrt(b^2 - 4ac))/2a
which is also a rational number.
Hence, if one root of a quadratic equation with rational coefficients is rational, then the other root must also be rational. Therefore, the correct answer is option C.
When a quadratic equation has rational coefficients, it can have roots that are either rational or irrational. Let's consider a quadratic equation:
ax^2 + bx + c = 0
If one of the roots is rational, say p/q, where p and q are integers with no common factors, then the other root must be of the form:
(-b + sqrt(b^2 - 4ac))/2a or (-b - sqrt(b^2 - 4ac))/2a
Let's assume that (-b + sqrt(b^2 - 4ac))/2a is the other root. We know that the sum of the roots is -b/a and the product of the roots is c/a. Therefore,
(-b + p/q) + (-b + sqrt(b^2 - 4ac))/2a = -b/a
Simplifying the above equation, we get:
sqrt(b^2 - 4ac) = (2ap - bq)/q
Since p/q is rational and a, b, c are rational coefficients, (2ap - bq)/q must be a rational number. Therefore, sqrt(b^2 - 4ac) must be a rational number as well. This implies that the other root must be of the form:
(-b - sqrt(b^2 - 4ac))/2a
which is also a rational number.
Hence, if one root of a quadratic equation with rational coefficients is rational, then the other root must also be rational. Therefore, the correct answer is option C.
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If one of the root of a quadratic equation with rational coefficients is rational, then other root must bea)Imaginaryb)Irrationalc)Rationald)None of theseCorrect answer is option 'C'. Can you explain this answer?
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