Find quadratic equation which has the following roots(i) 5 and -2(ii)

1.	Find quadratic equation which has the following roots(i) 5 and -2(ii) 1 + 2i
Answer» (i) root α = 5 and β = – 2 Then, sum of roots = α + β = 5 – 2 ⇒ α + β = 3 and product of roots = αβ = 5 × (-2) ⇒ αβ = -10 Hence, required equation whose roots are 5 and -2. x² – (sum of roots) x + product of roots = 0 ⇒ x² – 3x + (-10) = 0 ⇒ x² – 3x – 10 = 0 Hence, required equation is x² – 3x – 10 = 0 whose roots are 5 and -2. (ii) Roots α = 1 + 2i and β = 1 – 2i Then, sum of roots α + β = 1 + 2i + 1 – 2i = 2 and product of roots αβ = (1 + 2i)(1 – 2i) = 1 – 4i² = 1 + 4 = 5 Hence, required equation whose roots are 1 + 2i and 1 – 2i, x² – (sum of roots) x + Product of roots = 0 ⇒ x² – 2x + 5 = 0. Remark: If one root is 1 + 2i, then second not will be 1 – 2i.

Find quadratic equation which has the following roots(i) 5 and -2(ii) 1 + 2i

Answer»

(i) root α = 5 and β = – 2

Then, sum of roots = α + β = 5 – 2

⇒ α + β = 3

and product of roots = αβ = 5 × (-2)

⇒ αβ = -10

Hence, required equation whose roots are 5 and -2.

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

⇒ x² – 3x + (-10) = 0

⇒ x² – 3x – 10 = 0

Hence, required equation is x² – 3x – 10 = 0 whose roots are 5 and -2.

(ii) Roots α = 1 + 2i and β = 1 – 2i

Then, sum of roots α + β = 1 + 2i + 1 – 2i = 2

and product of roots αβ = (1 + 2i)(1 – 2i) = 1 – 4i² = 1 + 4 = 5

Hence, required equation whose roots are 1 + 2i and 1 – 2i,

x² – (sum of roots) x + Product of roots = 0

⇒ x² – 2x + 5 = 0.

Remark: If one root is 1 + 2i, then second not will be 1 – 2i.