Complete the following activity to form a quadratic equation. Activity:

The standard form off a quadratic equation is x 2 - ( α + β ) x + α β = 0 , where α , β are roots of equation. Here, given roots are 3 and 4 . Sum of the roots 3 + 4 = 7 Product of the root 3 × 4 = 12 So, if we construct a quadratic equation by given roots, we get x 2 - ( 3 + 4 ) x + 3 × 4 = 0 ⇒ x 2 - 7 x + 12 = 0 Hence, the quadratic equation with roots 3 and 4 is x 2 - 7 x + 12 = 0

If α + β = 5 and α 3 + β 3 = 35 , find the quadratic equation whose roots are α and β .

Given that, the roots of the required quadratic equation are α , β Also, given, α + β = 5 and α 3 + β 3 = 35 Now, α 3 + β 3 = ( α + β ) 3 - 3 α β ( α + β ) ⇒ 35 = ( 5 ) 3 - 3 α β ( 5 ) ⇒ 35 = 125 - 15 α β ⇒ 15 α β = 125 - 35 ⇒ 15 α β = 90 ⇒ α β = 90 15 ∴ α β = 6 We know that, the quadratic equation whose roots are α and β is, x 2 - ( α + β ) x + α β = 0 ⇒ x 2 - 5 x - 6 = 0 ∵ α + β = 5 , α β = 6 Hence, the required quadratic equation is x 2 - 5 x + 6 = 0 .