Question

Determine the quadratic equations, whose sum and product of roots are 

(i) -9, 20 

(ii) 5/3, 4 

(iii) -3/2, -1 

(iv) -(2 – a)², (a + 5)²

Answer 1

If the roots are given, general form of the quadratic equation is x – (sum of the roots) x² + product of the roots = 0. 

(i) Sum of the roots = -9

Product of the roots = 20 

The equation = x² – (-9x) + 20 = 0 

⇒ x² + 9x + 20 = 0

(ii) Sum of the roots = 5/3

Product of the roots = 4 

Required equation = x² – (sum of the roots)x + product of the roots

= 0 

⇒ x² – (5/3)x + 4 = 0 

⇒ 3x² – 5x + 12 = 0

(iii) Sum of the roots = (-3/2) 

(α + β) = -3/2

Product of the roots (αβ) = (-1) 

Required equation = x² – (α + β)x + αβ = 0 

x² – (-3/2)x – 1 = 0 

2x² + 3x – 2 = 0

(iv) α + β = – (2 – a)² 

αβ = (a + 5)² 

Required equation = x² – (α + β)x – αβ = 0 

⇒ x² – (-(2 – a)²)x + (a + 5)² = 0 

⇒ x² + (2 – a)² x + (a + 5)² = 0