Formation of Quadratic Equations With Given Roots

Formation of Quadratic Equations With Given Roots:

Let α and β be the roots of the equation ax2 + bx + c = 0, where a ≠ 0. Then, α + β = -b/a and αβ = c/a.

Now, the equation can be written as-

x2 +(b/a)x + c/a = 0 or, x2 – (-b/a)x + c/a = 0

⇒ x2 – (α + β)x + αβ = 0 ……….(i)

⇒ x2 – (sum of the roots). x + product of the roots = 0

Thus, the equation whose roots are α and β may be written as x2 – (α + β)x + αβ = 0.

Equation (i) can also be written as-

x2 – αx – βx + αβ = 0

⇒ x (x – α) – β (x – α) = 0

⇒ (x – α) (x – β) = 0

Thus, the equation with α and β as roots can also be written as (x – α) (x – β) = 0.

Example- If α, β are the roots of the equation x2 – 2x + 5 = 0 then form an equation whose roots are (i) 2α, 2β (ii) 1/α, 1/β (iii) α2, β2

Solution- The given equation is,
x2 – 2x + 5 = 0

Now, α and β are its roots.
∴ α + β = 2
and αβ = 5

(i) Now the roots of the required equation are 2α, 2β
∴ Sum of roots = 2α + 2β = 2 (α + β) = 2(2) = 4
and Product of roots = (2α) (2β) = 4αβ = 4(5) = 20

The required equation is,
x2 – 4x + 20 = 0

(ii) Now the roots of the required equation are 1/α, 1/β
∴ Sum of roots = 1/α + 1/β = (β + α)/αβ = 2/5
and Product of roots = (1/α) (1/β) = 1/αβ = 1/5

The required equation is,
x2 – (2/5)x + 1/5 = 0
5x2 – 2x + 1 = 0

(iii) Now the roots of the required equation are α2, β2
∴ Sum of roots = α2+ β2 = (α + β)2 – 2αβ = (2)2 – 2(5) = 4 – 10 = -6
and Product of roots = α2x β2 = α2β2 = (5)2 = 25

The required equation is,
x2 + 6x + 25 = 0

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