The quadratic equation is in the form of
ax^{2} + bx + c = 0
If α, β, are the roots of the quadratic equation, then the form of the quadratic equation as
x^{2} – (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
x^{2} – (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 + 20/15 α + β = 26/15 |
αβ = 2/5 × 4/3 αβ = 8/15 |
x^{2} – (26/15)x + 8/15 = 0
15x^{2} – 26x + 8 = 0
Problem 2 :
Roots : 2/3 and 5/6
Solution :
Roots : 2/3 and 5/6
α = 2/3, β = 5/6
x^{2} – (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 = 9/6 α + β = 3/2 |
αβ = 2/3 × 5/6 = 10/18 αβ = 5/9 |
x^{2} – (3/2)x + 5/9 = 0
x^{2}/1 × (18/18) – 3/2 × (9/9) + 5/9 × (2/2)
18x^{2} – 27x + 10 = 0
Problem 3 :
Roots : (3 + √5) and (3 - √5)
Solution :
Roots : (3 + √5) and (3 - √5)
α = (3 + √5), β = (3 - √5)
x^{2} – (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 |
x^{2} – 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)
x^{2} – (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 |
x^{2} – 4x - 14 = 0
Problem 5 :
Roots : (3 + 4i) and (3 – 4i)
Solution :
Roots : (3 + 4i) and (3 – 4i)
α = (3 + 4i), β = (3 – 4i)
x^{2} – (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 |
x^{2} – (5 + 6i)x + 0 = 0
Problem 7 :
One root of 4 + √7
Solution :
(4 + √7)
α = (4 + √7), β = 0
x^{2} – (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 |
x^{2} – (4 + √7)x + 0 = 0
May 21, 24 08:51 PM
May 21, 24 08:51 AM
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