Find the roots of the equation 3 x 2 + 10 x - 8 3 = 0 .

Given, 3 x 2 + 10 x - 8 3 = 0 The roots of the quadratic equation a x 2 + b x + c are x = - b ± b 2 - 4 a c 2 a Then, the roots of 3 x 2 + 10 x - 8 3 = 0 are x = - 10 ± 10 2 + 4 8 ( 3 ) 2 · 3 ⇒ x = - 10 ± 100 + 32 ( 3 ) 2 3 ⇒ x = - 10 ± 196 2 3 ⇒ x = - 10 ± 14 2 3 ⇒ x = - 10 + 14 2 3 , - 10 - 14 2 3 ⇒ x = 2 3 , - 12 3 ⇒ x = 2 3 , - 4 3 ∴ The roots are 2 3 , - 4 3 .

Find the roots of the equation 6 5 x 2 - 9 x - 3 5 = 0 .

Given, 6 5 x 2 - 9 x - 3 5 = 0 The roots of the quadratic equation a x 2 + b x + c are x = - b ± b 2 - 4 a c 2 a Then, the roots of 6 5 x 2 - 9 x - 3 5 = 0 are x = 9 ± - 9 2 + 4 6 5 ( 3 5 ) 2 · 6 5 ⇒ x = 9 ± 81 + 360 12 5 ⇒ x = 9 ± 441 12 5 ⇒ x = 9 ± 21 12 5 ⇒ x = 9 + 21 12 5 , 9 - 21 12 5 ⇒ x = 30 12 5 , - 12 12 5 ⇒ x = 5 2 5 , - 1 5 ⇒ x = 5 2 , - 1 5 ∴ The roots are 5 2 , - 1 5 .

Form quadratic equation whose roots are m n , - n m m ≠ 0 , n ≠ 0 .

Let, α = m n and β = - n m Then, α + β = m n - n m and α β = m n - n m ⇒ α + β = m 2 - n 2 m n and α β = - 1 The quadratic equation is of the form x 2 - sum of roots x + product of roots = 0 ⇒ x 2 - ( α + β ) x + α β = 0 ⇒ x 2 - m 2 - n 2 m n x - 1 = 0 ⇒ m n x 2 - m 2 - n 2 x - m n = 0 ∴ The required equation is m n x 2 + n 2 - m 2 x - m n = 0 .

Form quadratic equation whose roots are p - q p + q , - p + q p - q p ≠ ± q .

Let, α = p - q p + q and β = - p + q p - q Then, α + β = p - q p + q - p + q p - q and α β = p - q p + q · - p + q p - q ⇒ α + β = p - q p - q - p + q p + q p + q p - q and α β = - 1 ⇒ α + β = p 2 - 2 p q + q 2 - p 2 - 2 p q - q 2 p 2 - q 2 and α β = - 1 ⇒ α + β = - 4 p q p 2 - q 2 and α β = - 1 The quadratic equation is of the form x 2 - sum of roots x + product of roots = 0 ⇒ x 2 - ( α + β ) x + α β = 0 ⇒ x 2 - - 4 p q p 2 - q 2 x - 1 = 0 ⇒ p 2 - q 2 x 2 + 4 p q x - p 2 - q 2 = 0 ∴ The required equation is p 2 - q 2 x 2 + 4 p q x - p 2 + q 2 = 0 .

Form quadratic equation whose roots are 7 ± 2 5 .

Let, α = 7 + 2 5 and β = 7 - 2 5 Then, α + β = 7 + 2 5 + 7 - 2 5 and α β = 7 + 2 5 · 7 - 2 5 ⇒ α + β = 14 and α β = 7 2 - 2 5 2 ⇒ α + β = 14 and α β = 49 - 20 ⇒ α + β = 14 and α β = 29 The quadratic equation is of the form x 2 - sum of roots x + product of roots = 0 ⇒ x 2 - ( α + β ) x + α β = 0 ⇒ x 2 - 14 x + 29 = 0 ∴ The required equation is x 2 - 14 x + 29 = 0 .