First slide

Theory of equations

Question

If $α, β$ are the roots of the equation $x^{2} - 3 x + 5 = 0$ and $γ, δ$ are the roots of the equation $x^{2} + 5 x - 3 = 0$ , then the equation whose roots are $αγ + βδ and αδ + βγ$ is

Moderate

A

$x^{2} - 15 x - 158 = 0$
B

$x^{2} + 15 x - 158 = 0$
C

$x^{2} - 15 x + 158 = 0$
D

$x^{2} + 15 x + 158 = 0$

Solution

$∵ α + β = 3, αβ = 5, γ + δ = (- 5), γδ = (- 3)$
Sum of roots
$\begin{aligned} = (αγ + βδ) + (αδ + βγ) \\ = (α + β) (γ + δ) = 3 \times (- 5) = (- 15) \end{aligned}$
Product of roots
$\begin{aligned} = (αγ + βδ) (αδ + βγ) \\ = α^{2} γδ + {αβγ}^{2} + {βαδ}^{2} + β^{2} γδ \\ = γδ (α^{2} + β^{2}) + αβ (γ^{2} + δ^{2}) \end{aligned}$
$\begin{aligned} = - 3 (α^{2} + β^{2}) + 5 (γ^{2} + δ^{2}) \\ = - 3 [(α + β)^{2} - 2 αβ] + 5 [(γ + δ)^{2} - 2 γδ] \\ = - 3 [9 - 10] + 5 [25 + 6] = 158 \end{aligned}$
Required equation is $x^{2} + 15 x + 158 = 0$

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