pls answer this all! thank you!
Answers
Step-by-step explanation:
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 the second query, as we've proved √5 irrational. Therefore 2-√5 is also irrational because difference of a rational and an irrational number is always an irrational number.
For a quadratic equation, ax
2
+bx+c=0,
The sum of the roots=
a
−b
and product of the roots is
a
c
Here sum of roots= α+β=
2
−5
Product of roots=αβ=
2
1
Therefore,α+β+αβ=
2
−5
+
2
1
⇒ α+β+αβ=−2Answer. a2 - b2 - a - b = [(a - b)(a + b)] - (a + b) = (a + b)(a - b - 1). Therefore, factors of a2 - b2 - a - b are (a + b)(a - b - 1Anything is possible through careful inspection. Here is a star with numbers printed all over it, find the number that should go into the empty space. Complete The Star by filling in the number.
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hope it helps you...remaining solutionsI will post as a question..and make sure with those answers...Thank you
