`alpha ` and `beta ` are the roots of `x^(2) - 5x - 1 = 0 ` . Complete the following activity to find the value of `x^(2) + beta^(3)`.

Explanation

`x^(2) - 5 x - 1= 0` , Here, ` a = 1 ,b = square , c = - 1 `
`alpha + beta = ( - b)/( a ) = ( - ( -5))/( 1) = 5, alpha beta = ( c )/( a) = square = square `
`alpha^(2) + beta ^(3) = ( alpha + beta) ^(3) - square `
` = ( 5)^(3) - 3 xx ( - 1) xx ( 5) `
` = 125 + square = square `
Activity `: a = 1 , b = -5, c = - 1 `
`alpha + beta = ( - b)/( a) = (-(-5))/( 1) = 5, alpha beta = ( c )/( a) = ( - 1)/( 1) = - 1 `
` alpha^(3) + beta^(2) = ( alpha + beta )^(3) - 3alpha beta ( alpha + beta)`
` = ( 5)^(3) - 3 xx ( -1) xx ( 5)`
`= 125 + 15 = 140`