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If alpha and beta are root of equation 2x^2-5x 1=0 then find alpha^2 +beta^2 and alpha^3+ beta^3?
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If alpha and beta are root of equation 2x^2-5x 1=0 then find alpha^2 +...
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If alpha and beta are root of equation 2x^2-5x 1=0 then find alpha^2 +...
Solution:
Given equation is 2x^2-5x+1=0.
We can find the values of alpha and beta using the quadratic formula.
Finding Alpha and Beta:
Using the quadratic formula, we get
alpha = (5 + sqrt(5))/4 and beta = (5 - sqrt(5))/4.
Finding Alpha^2 and Beta^2:
We can find alpha^2 and beta^2 as follows:
alpha^2 = [(5 + sqrt(5))/4]^2 = (25 + 10sqrt(5) + 5)/16 = (30 + 10sqrt(5))/16
beta^2 = [(5 - sqrt(5))/4]^2 = (25 - 10sqrt(5) + 5)/16 = (30 - 10sqrt(5))/16
Finding Alpha^3 and Beta^3:
We can use the identity (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 to find alpha^3 and beta^3.
alpha^3 = [(5 + sqrt(5))/4]^3 = (125 + 75sqrt(5) + 15sqrt(5) + 5sqrt(5)^2)/64
= (125 + 90sqrt(5))/64
beta^3 = [(5 - sqrt(5))/4]^3 = (125 - 75sqrt(5) + 15sqrt(5) - 5sqrt(5)^2)/64
= (125 - 90sqrt(5))/64
Therefore, alpha^2 beta^2 = [(30 + 10sqrt(5))/16][(30 - 10sqrt(5))/16] = (900 - 500)/256 = 2.34375.
Similarly, alpha^3 beta^3 = [(125 + 90sqrt(5))/64][(125 - 90sqrt(5))/64] = (15625 - 10125)/4096 = 1.3652.
Hence, we have found alpha^2 beta^2 and alpha^3 beta^3.
Given equation is 2x^2-5x+1=0.
We can find the values of alpha and beta using the quadratic formula.
Finding Alpha and Beta:
Using the quadratic formula, we get
alpha = (5 + sqrt(5))/4 and beta = (5 - sqrt(5))/4.
Finding Alpha^2 and Beta^2:
We can find alpha^2 and beta^2 as follows:
alpha^2 = [(5 + sqrt(5))/4]^2 = (25 + 10sqrt(5) + 5)/16 = (30 + 10sqrt(5))/16
beta^2 = [(5 - sqrt(5))/4]^2 = (25 - 10sqrt(5) + 5)/16 = (30 - 10sqrt(5))/16
Finding Alpha^3 and Beta^3:
We can use the identity (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 to find alpha^3 and beta^3.
alpha^3 = [(5 + sqrt(5))/4]^3 = (125 + 75sqrt(5) + 15sqrt(5) + 5sqrt(5)^2)/64
= (125 + 90sqrt(5))/64
beta^3 = [(5 - sqrt(5))/4]^3 = (125 - 75sqrt(5) + 15sqrt(5) - 5sqrt(5)^2)/64
= (125 - 90sqrt(5))/64
Therefore, alpha^2 beta^2 = [(30 + 10sqrt(5))/16][(30 - 10sqrt(5))/16] = (900 - 500)/256 = 2.34375.
Similarly, alpha^3 beta^3 = [(125 + 90sqrt(5))/64][(125 - 90sqrt(5))/64] = (15625 - 10125)/4096 = 1.3652.
Hence, we have found alpha^2 beta^2 and alpha^3 beta^3.
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If alpha and beta are root of equation 2x^2-5x 1=0 then find alpha^2 +beta^2 and alpha^3+ beta^3?
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If alpha and beta are root of equation 2x^2-5x 1=0 then find alpha^2 +beta^2 and alpha^3+ beta^3? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If alpha and beta are root of equation 2x^2-5x 1=0 then find alpha^2 +beta^2 and alpha^3+ beta^3? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If alpha and beta are root of equation 2x^2-5x 1=0 then find alpha^2 +beta^2 and alpha^3+ beta^3?.
If alpha and beta are root of equation 2x^2-5x 1=0 then find alpha^2 +beta^2 and alpha^3+ beta^3? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If alpha and beta are root of equation 2x^2-5x 1=0 then find alpha^2 +beta^2 and alpha^3+ beta^3? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If alpha and beta are root of equation 2x^2-5x 1=0 then find alpha^2 +beta^2 and alpha^3+ beta^3?.
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