Finding quadratic equation with given sum and product of roots, conditions for equal roots, real roots and no root

Finding Quadratic Equation Given the Sum and Product of Roots – 6 to 10

Find the sum and product of the roots of each of the following quadratic equations:

(a) 2x² + 3x – 1 = 0

(b) 3x² – 5x – 2 = 0

(c) x² – 4x – 3 = 0

(d) ½ x² – 3x – 1 = 0

Solution

(a) 2x² + 3x – 1 = 0

a = 2; b = 3; c = -1

Let α and β be the roots of the equation, then

α + β= -b/a = -3/2

α β = c/a = -1/2

(b) 3x² – 5x – 2 = 0

a = 3; b = -5; c = -2

Let α and β be the root of the equation, then

α + β = -b/a = 5/3

α β = c/a = -3

(c) x² – 4x – 3 = 0

a = 1; b = 4; c = -3

Let α and β be the root of the equation, then

α + β = -b/a = 4/1

α β = c/a = -3

(d) ½ x² – 3x – 1 = 0

a = ½, b = -3, c = -1

Let α and β be the root of the equation, then

α + β = -b/a = 4/1

α β = c/a = -1/(1/2) = -2

Symmetric Properties of Roots

Theory

(1) Find the constants p, q and r such that 3x² – 12x + 16 = p (x + q)² + r

(2) If α and β are the roots of x² – 10x + 2 = 0, find α³ – β³.