Find the quadratic equation whose roots are 4 5 and - 3 5 .

We know that if the roots of a quadratic equation are α and β , then the quadratic equation will be x 2 - α + β x + α β = 0 . So, x 2 - Sum of the roots x + Products of root = 0 Here, roots are 4 5 and - 3 5 . Thus, we get x 2 - 4 5 + - 3 5 x + 4 5 × - 3 5 = 0 ⇒ x 2 - 4 5 - 3 5 x + - 12 25 = 0 ⇒ x 2 - 1 5 x - 12 25 = 0 Hence, the required quadratic equation is x 2 - 1 5 x - 12 25 = 0 .

Find the quadratic equation whose roots are - 1 3 and 2 5 .

We know that if the roots of a quadratic equation are α and β , then the quadratic equation will be x 2 - α + β x + α β = 0 . So, x 2 - Sum of the roots x + Products of root = 0 Here, roots are - 1 3 and 2 5 . Thus, we get x 2 - - 1 3 + 2 5 x + - 1 3 × 2 5 = 0 ⇒ x 2 - - 5 + 6 15 x + - 2 15 = 0 ⇒ x 2 - x 15 - 2 15 = 0 Hence, the required quadratic equation is x 2 - x 15 - 2 15 = 0 .

Find the quadratic equation whose roots are 3 - 7 and 3 + 7 .

We know that if the roots of a quadratic equation are α and β , then the quadratic equation will be x 2 - α + β x + α β = 0 . So, x 2 - Sum of the roots x + Products of roots = 0 Here, roots are 3 - 7 and 3 + 7 . Thus, we get x 2 - 3 - 7 + 3 + 7 x + 3 - 7 × 3 + 7 = 0 ⇒ x 2 - 2 3 x - 46 = 0 Hence, the required quadratic equation is x 2 - 2 3 x - 46 = 0 .

Find the quadratic equation whose roots are 6 + 5 and 6 - 5 .

We know that if the roots of a quadratic equation are α and β , then the quadratic equation will be x 2 - α + β x + α β = 0 . So, x 2 - Sum of the roots x + Products of root = 0 Here, roots are 6 + 5 and 6 - 5 . Thus, we get x 2 - 6 + 5 + 6 - 5 x + 6 + 5 × 6 - 5 = 0 ⇒ x 2 - 12 x + 31 = 0 Hence, the required quadratic equation is x 2 - 12 x + 31 = 0 .