Question

if alpha and beta are the zeroes of quadratic polynomial f(x)=ax^2+bx+c then evaluate alpha^2beta+alpha beta^2,alpha^2/beta+beta^2/alpha,1/alpha+1/beta-2alpha beta? please answer.

Answer 1

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Answer 2

GIVEN :

Alpha , beta are the zeroes of f(x)=ax^2+bx+c .

TO FIND :

Evaluate,

◆ alpha^2beta+alpha beta^2,

◆alpha^2/beta+beta^2/alpha,

◆1/alpha+1/beta-2alpha beta

SOLUTION :

◆ We know , roots of solution

Alpha (A) , Beta (B) can be written as,

◆Product of roots ,AB = c/a

◆Sum of roots , A+B = -b/a

Solving ,

◆A^2B + B^2A = AB (A+ B)

= c/a × -b/a = -bc/a^2.

◆A^2/B + B^2/A = (A^3 +B^3) /AB

= (A+B)(A^2 - AB + B^2)/AB

= (A+B){(A+B)^2 -2AB-AB }/AB

Substituting values,

= (-b/a^2c) (b^2-3ac)

◆1/A + 1/B + 2AB = A +B /AB + 2AB

={( -b/a )÷ c/a} + 2c/a

= (-ab -2c^2 )/ac.

ANSWER :

◆A^2B + B^2A = -bc/a^2.

◆A^2/B + B^2/A =(-b/a^2c) (b^2-3ac)

◆1/A + 1/B + 2AB = (-ab -2c^2 )/ac.