FIND A QUADRATIC EQUATION WITH INTEGRAL COEFFICIENTS GIVEN ROOTS

Write the quadratic equation with Integral coefficients which have the following roots :

Problem 1 :

Roots : 2/5 and 4/3

Solution :

Roots : 2/5 and 4/3

α = 2/5, β = 4/3

x² – (α + β)x + αβ = 0

α + β = sum of roots

αβ = product of roots

x² – (26/15)x + 8/15 = 0

15x² – 26x + 8 = 0

Problem 2 :

Roots : 2/3 and 5/6

Solution :

Roots : 2/3 and 5/6

α = 2/3, β = 5/6

x² – (α + β)x + αβ = 0

α + β = sum of roots

αβ = product of roots

x² – (3/2)x + 5/9 = 0

x²/1 × (18/18) – 3/2 × (9/9) + 5/9 × (2/2)

18x² – 27x + 10 = 0

Problem 3 :

Roots : (3 + √5) and (3 - √5)

Solution :

Roots : (3 + √5) and (3 - √5)

α = (3 + √5), β = (3 - √5)

x² – (α + β)x + αβ = 0

α + β = sum of roots

αβ = product of roots

x² – 6x + 4 = 0

Problem 4 :

Roots : (2 + 3√2) and (2 - 3√2)

Solution :

Roots : (2 + 3√2) and (2 - 3√2)

α = (2 + 3√2), β = (2 - 3√2)

x² – (α + β)x + αβ = 0

α + β = sum of roots

αβ = product of roots

x² – 4x - 14 = 0

Problem 5 :

Roots : (3 + 4i) and (3 – 4i)

Solution :

Roots : (3 + 4i) and (3 – 4i)

α = (3 + 4i), β = (3 – 4i)

x² – (α + β)x + αβ = 0

α + β = sum of roots

αβ = product of roots

 x² – (5 + 6i)x + 0 = 0

Problem 7 :

One root of 4 + √7

Solution :

(4 + √7)

α = (4 + √7), β = 0

x² – (α + β)x + αβ = 0

α + β = sum of roots

αβ = product of roots

x² – (4 + √7)x + 0 = 0