First slide

Theory of equations

Question

If $α$ and $β$ are the roots of equation 2x² - 5x + 7 = 0, then the equation whose roots are 2 $α$ + 3 $β$ and 3 $β$ + 2 $β$ , is

Moderate

A

2x² - 25x+ 82 = 0
B

2x² + 25x+ 82 = 0
C

x² - 25x+ 82 = 0
D

None of these

Solution

Since, $α$ and $β$ be the roots of the equation
2x² -5x+ 7 =0,then
$α + β = \frac{5}{2} and αβ = \frac{7}{2}$
Now sum of roots = $(2 α + 3 β) + (3 α + 2 β) = 5 (α + β) = (\frac{25}{2})$
and product of roots
$= (2 α + 3 β) (3 α + 2 β)$
$= 6 (α^{2} + β^{2}) + 13 αβ$
$\begin{array}{l} = 6 [(α + β)^{2} - 2 αβ] + 13 αβ \\ = [6 \times (\frac{25}{4} - 7) + \frac{91}{2}] = 41 \end{array}$
The required equation is
$\begin{array}{l} x^{2} - \frac{25}{2} x + 41 = 0 \\ \Rightarrow 2 x^{2} - 25 x + 82 = 0 \end{array}$

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