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Ask QuestionPosted by Sudhanshu Raj 6 years, 4 months ago
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Posted by Rajandar Kumar 6 years, 4 months ago
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Massenger Bhardwaj 6 years, 4 months ago
Posted by Sachin Kumar 6 years, 4 months ago
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Gaurav Seth 6 years, 4 months ago
Let's prove this by the method of contradiction-
Say, √5 is a rational number. ∴ It can be expressed in the form p/q where p,q are co-prime integers.
⇒√5=p/q
⇒5=p²/q² {Squaring both the sides}
⇒5q²=p² (1)
⇒p² is a multiple of 5. {Euclid's Division Lemma}
⇒p is also a multiple of 5. {Fundamental Theorm of arithmetic}
⇒p=5m
⇒p²=25m² (2)
From equations (1) and (2), we get,
5q²=25m²
⇒q²=5m²
⇒q² is a multiple of 5. {Euclid's Division Lemma}
⇒q is a multiple of 5.{Fundamental Theorm of Arithmetic}
Hence, p,q have a common factor 5. this contradicts that they are co-primes. Therefore, p/q is not a rational number. This proves that √5 is an irrational number.
For you second query, as we've proved √5 irrational. Therefore √5+3 is also irrational because sum of a rational and an irrational number is always an irrational number.
Posted by King King 6 years, 4 months ago
- 3 answers
Gaurav Seth 6 years, 4 months ago
Given in the question the equation:
3x² - 10x + k = 0
suppose 1st root of this equation is r
so as stated in the question, second root of this equation will be 1/r
Adding the roots gives = −b/a
Multiplying the roots gives = c/a
a b c
3x² - 10x + k = 0
r x 1 / r = c / a
r / r = k / 3
1 = k / 3
3 = k
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Aadya Singh ? 6 years, 4 months ago
1Thank You