First slide

Theory of equations

Question

$α, β are the roots of the equation x^{2} - 2 x + 3 = 0 . Then the equation whose roots are$ $P = α^{3} - 3 α^{2} + 5 α - 2 and Q = β^{3} - β^{2} + β + 5 is$

Moderate

A

$x^{2} + 3 x + 2 = 0$
B

$x^{2} - 3 x - 2 = 0$
C

$x^{2} - 3 x + 2 = 0$
D

None of these

Solution

$Given α, β are roots of equation x^{2} - 2 x + 3 = 0$
$\Rightarrow α^{2} - 2 α + 3 = 0 - - - - - (1)$
$3 = - α^{2} + 2 α - - - - - (2)$
from eq1
$\begin{array}{ll} ∴ α^{2} = 2 α - 3 \\ m u l t i p l y b y α w e g e t α^{3} = 2 α^{2} - 3 α \\ s u b f r o m e q 2 \\ ∴ P = (2 α^{2} - 3 α) - 3 α^{2} + 5 α - 2 \\ = - α^{2} + 2 α - 2 = 3 - 2 = 1 \end{array}$
$and β^{2} - 2 β + 3 = 0 - - - - (3)$
$β^{3} - 2 β^{2} + 3 β = 0$
$β^{3} = 2 β^{2} - 3 β - - - - (4)$
$S i m i l a r l y Q = β^{3} - β^{2} + β + 5$
sub from eq 4
$Q = 2 β^{2} - 3 β - β^{2} + β + 5$
$= β^{2} - 2 β + 5$
=-3+5=2 from eq 3
P=1 , Q=2
So, sum of roots = 3, and product of roots = 2.
$Then required equation is x^{2} - 3 x + 2 = 0 .$

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