# FIND A QUADRATIC EQUATION WITH INTEGRAL COEFFICIENTS GIVEN ROOTS

The quadratic equation is in the form of

ax2 + bx + c = 0

If α, β, are the roots of the quadratic equation, then the form of the quadratic equation as

x2 – (α + β)x + αβ = 0

Where,

α + β = sum of roots

αβ = product of 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

 α + β = 2/5 + 4/3= 2/5 × (3/3) + 4/3 × (5/5)= 6/15 + 20/15α + β = 26/15 αβ = 2/5 × 4/3αβ = 8/15

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

 α + β = 2/3 + 5/6= 2/3 × (2/2) + 5/6= 4/6 + 5/6= 9/6α + β = 3/2 αβ = 2/3 × 5/6= 10/18αβ = 5/9

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

 α + β = (3 + √5) + (3 - √5)= 3 + √5 + 3 - √5 α + β = 6 αβ = (3 + √5) × (3 - √5)= 9 - 3√5 + 3√5 – 5= 4

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

 α + β = (2 + 3√2) + (2 - 3√2)= 2 + 3√2 + 2 - 3√2= 4 αβ = (2 + 3√2) × (2 - 3√2)= 2(2 - 3√2) + 3√2(2 - 3√2)= 4 – 2(3√2) + 2(3√2) – (3√2)(3√2)= 4 – 18= -14

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

 α + β = (5 + 6i) + 0= 5 + 6i αβ = (5 + 6i) × 0= 0

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

 α + β = (4 + √7) + 0= 4 + √7 αβ =  (4 + √7) × 0= 0

x2 – (4 + √7)x + 0 = 0

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