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  4. The roots of the equation |1&4&20 1&-2&5 1&2x&5x2| = 0 are :

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Q. The roots of the equation $\begin{vmatrix}1&4&20\\ 1&-2&5\\ 1&2x&5x^{2}\end{vmatrix} = 0 $ are :

Determinants

Solution:

Given $\begin{vmatrix}1&4&20\\ 1&-2&5\\ 1&2x&5x^{2}\end{vmatrix} = 0 $
Operate $R_2 \to R_2 -R_1$ and $R_3 \to R_3 - R_1$
Thus we get: $\begin{vmatrix}1&4&20\\ 0& - 6 & -15\\ 0& 2x - 4 &5x^2 - 20 \end{vmatrix} = 0 $
$\Rightarrow 1[- 30x^2 + 120 + 30x - 60] = 0 $
$\Rightarrow 30x^2 - 30x - 60 = 0 $
$\Rightarrow x = - 1, 2. $
$\therefore $ roots are - 1 and 2