If α , β are the roots of the equation x 2 + p x + q = 0 , q ≠ 0 then find the values (in terms of p and q ) ( α + 1 ) ( β + 1 )

Given, ( α + 1 ) ( β + 1 ) Since α and β are the roots of the equation x 2 + p x + q = 0 , We have α + β = - p 1 = - p and α β = q 1 = q So, ( α + 1 ) ( β + 1 ) = α β + α + β + 1 = q - p + 1 Thus, the value of ( α + 1 ) ( β + 1 ) is q - p + 1

If α , β are the roots of the equation x 2 + p x + q = 0 , q ≠ 0 then find the values (in terms of p and q ) ( α + 1 ) ( β + 1 )

Given, ( α + 1 ) ( β + 1 ) Since α and β are the roots of the equation x 2 + p x + q = 0 , We have α + β = - p 1 = - p and α β = q 1 = q So, ( α + 1 ) ( β + 1 ) = α β + α + β + 1 = q - p + 1 Thus, the value of ( α + 1 ) ( β + 1 ) is q - p + 1

If α , β are the roots of the equation x 2 + p x + q = 0 , q ≠ 0 then find the values (in terms of p and q ) 1 α 2 + 1 β 2

Given, 1 α 2 + 1 β 2 Since α and β are the roots of the equation x 2 + p x + q = 0 , We have α + β = - p 1 = - p and α β = q 1 = q So, 1 α 2 + 1 β 2 = β 2 + α 2 α 2 β 2 = α + β 2 - 2 α β α β × α β = - p 2 - 2 q q = p 2 - 2 q q

If α , β are the roots of the equation x 2 + p x + q = 0 , q ≠ 0 then find the values (in terms of p and q ) α 3 + β 3

Given, α 3 + β 3 Since α and β are the roots of the equation x 2 + p x + q = 0 , We have α + β = - p 1 = - p and α β = q 1 = q So, α 3 + β 3 = α + β 3 + 3 α β ( α + β ) = - p 3 - 3 p q

If α , β are the roots of the equation x 2 + p x + q = 0 , q ≠ 0 then find the values (in terms of p and q ) α 2 β + β 2 α

Given, α 2 β + β 2 α Since α and β are the roots of the equation x 2 + p x + q = 0 , We have α + β = - p 1 = - p and α β = q 1 = q So, α 2 β + β 2 α = α 3 + β 3 β α = ( α + β ) 3 - 3 α β ( α + β ) β α = - p 3 + 3 p q q

If α , β are the roots of the equation x 2 + p x + q = 0 , q ≠ 0 then find the values (in terms of p and q ) ( α 2 + 2 ) ( β 2 + 2 )

Given, ( α 2 + 2 ) ( β 2 + 2 ) Since α and β are the roots of the equation x 2 + p x + q = 0 , We have α + β = - p 1 = - p and α β = q 1 = q So, ( α 2 + 2 ) ( β 2 + 2 ) = α 2 β 2 + 2 α 2 + 2 β 2 + 4 = q 2 + 2 ( α 2 + β 2 ) + 4 = q 2 + 2 ( α + β ) 2 - 2 α β + 4 = q 2 + 2 ( - p ) 2 - 2 q + 4 = q 2 + 2 p 2 - 2 q + 4

If α , β are the roots of the equation x 2 + p x + q = 0 , q ≠ 0 then form the quadratic equation whose roots are 3 α , 3 β

Given, 3 α , 3 β Since α and β are the roots of the equation x 2 + p x + q = 0 , We have α + β = - p 1 = - p and α β = q 1 = q So, x 2 + 3 α + 3 β x + 3 α × 3 β = 0 ⇒ x 2 + 3 ( - p ) x + 9 q = 0 ⇒ x 2 - 3 p x + 9 q = 0

If α , β are the roots of the equation x 2 + p x + q = 0 , q ≠ 0 then form the quadratic equation whose roots are α 2 , β 2

Given, α 2 , β 2 Since α and β are the roots of the equation x 2 + p x + q = 0 , We have α + β = - p 1 = - p and α β = q 1 = q So, ⇒ x 2 + α 2 + β 2 x + α 2 × β 2 = 0 ⇒ x 2 + α + β 2 x + α β 4 = 0 ⇒ 4 x 2 - 2 p x + q = 0 Multiplying both sides by 4 Hence, the required quadratic equation is 4 x 2 - 2 p x + q = 0 .

If α , β are the roots of the equation x 2 + p x + q = 0 , q ≠ 0 then form the quadratic equation whose roots are α - 1 , β - 1

Given, α - 1 , β - 1 Since α and β are the roots of the equation x 2 + p x + q = 0 , We have α + β = - p 1 = - p and α β = q 1 = q So, x 2 + α - 1 + β - 1 x + α - 1 β - 1 = 0 x 2 + α + β - 2 x + ( α β - β - α + 1 ) = 0 x 2 + - q - 2 x + ( q - p + 1 ) = 0

If α , β are the roots of the equation x 2 + p x + q = 0 , q ≠ 0 then form the quadratic equation whose roots are α 2 , β 2

Given, α 2 , β 2 Since α and β are the roots of the equation x 2 + p x + q = 0 , We have α + β = - p 1 = - p and α β = q 1 = q So, x 2 + α 2 + β 2 x + α 2 β 2 = 0 ⇒ x 2 + α + β 2 x + q 2 = 0 ⇒ x 2 + α + β 2 + 2 α β x + q 2 = 0 ⇒ x 2 + - p 2 + 2 q x + q 2 = 0