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If alpha and beta are the zeroes of the quadratic polynomial ax^2+bx+c then calculate (i)alpha^2+beta^2 (ii) alpha/beta+beta/alpha (iii) 1/alpha^3+1/beta^3 (iv) alpha^2/beta+beta^2/alpha?
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If alpha and beta are the zeroes of the quadratic polynomial ax^2+bx+c...
Solution:
Given quadratic polynomial ax^2 + bx + c
Zeroes of the polynomial are α and β
Sum of zeroes = α + β = -b/a
Product of zeroes = αβ = c/a
(i) α^2β^2 = (αβ)^2 = (c/a)^2
(ii) α/β and β/α
α/β = αβ/β^2 = c/aβ
β/α = βα/α^2 = c/aα
(iii) 1/α^3 and 1/β^3
1/α^3 = 1/(α^2 α) = (αβ/a)^3
1/β^3 = 1/(β^2 β) = (αβ/a)^3
(iv) α^2/β and β^2/α
α^2/β = αβ/β^2 = c/aβ
β^2/α = βα/α^2 = c/aα
Given quadratic polynomial ax^2 + bx + c
Zeroes of the polynomial are α and β
Sum of zeroes = α + β = -b/a
Product of zeroes = αβ = c/a
(i) α^2β^2 = (αβ)^2 = (c/a)^2
(ii) α/β and β/α
α/β = αβ/β^2 = c/aβ
β/α = βα/α^2 = c/aα
(iii) 1/α^3 and 1/β^3
1/α^3 = 1/(α^2 α) = (αβ/a)^3
1/β^3 = 1/(β^2 β) = (αβ/a)^3
(iv) α^2/β and β^2/α
α^2/β = αβ/β^2 = c/aβ
β^2/α = βα/α^2 = c/aα
Community Answer
If alpha and beta are the zeroes of the quadratic polynomial ax^2+bx+c...
Given that ax^2+bx+c, by which we can get alpha+beta=(-b/a) and alpha*beta=(c/a) then (1) alpha^2+beta^2=(alpha+beta)^2 -2*alpha*beta=(b^2-2ac)/a^2 (2)alpha/beta+beta/alpha=(alpha^2+beta^2)/alpha*beta=[(b^2-2ac)/a^2]/c/a by solving it we get (b^2-2ac)/ac (3) 1/alpha^3+1/beta^3=(beta^3+alpha^3)/(alpha*beta)^3=[(alpha+beta)(alpha^2+beta^2-alpha*beta)]/(alpha*beta)^3 by putting the value of (alpha+beta),(alpha^2+beta^2)&(alpha*beta) we get the answer that is( -b^3+3abc)/c^3 (4) alpha^2/beta+beta^2/alpha=(alpha^3tbeta^3)/alpha*beta=[(alpha+beta)(alpha^2+beta^2-alpha*beta)]/(alpha*beta) again by putting the value of (alpha+beta),(alpha^2+beta^2)&(alpha*beta)as I already give in 1st question; we can get the answer which is (-b^3+3abc)/ac^2
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If alpha and beta are the zeroes of the quadratic polynomial ax^2+bx+c then calculate (i)alpha^2+beta^2 (ii) alpha/beta+beta/alpha (iii) 1/alpha^3+1/beta^3 (iv) alpha^2/beta+beta^2/alpha? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If alpha and beta are the zeroes of the quadratic polynomial ax^2+bx+c then calculate (i)alpha^2+beta^2 (ii) alpha/beta+beta/alpha (iii) 1/alpha^3+1/beta^3 (iv) alpha^2/beta+beta^2/alpha? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If alpha and beta are the zeroes of the quadratic polynomial ax^2+bx+c then calculate (i)alpha^2+beta^2 (ii) alpha/beta+beta/alpha (iii) 1/alpha^3+1/beta^3 (iv) alpha^2/beta+beta^2/alpha?.
If alpha and beta are the zeroes of the quadratic polynomial ax^2+bx+c then calculate (i)alpha^2+beta^2 (ii) alpha/beta+beta/alpha (iii) 1/alpha^3+1/beta^3 (iv) alpha^2/beta+beta^2/alpha? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If alpha and beta are the zeroes of the quadratic polynomial ax^2+bx+c then calculate (i)alpha^2+beta^2 (ii) alpha/beta+beta/alpha (iii) 1/alpha^3+1/beta^3 (iv) alpha^2/beta+beta^2/alpha? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If alpha and beta are the zeroes of the quadratic polynomial ax^2+bx+c then calculate (i)alpha^2+beta^2 (ii) alpha/beta+beta/alpha (iii) 1/alpha^3+1/beta^3 (iv) alpha^2/beta+beta^2/alpha?.
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