The equation $5 x^{2} + (9 - 4 k) x + 2 k^{2} = 0$ and $5 x + 9 = 0$ are satisfied by the same value of $x$ . Find the values of $k$ .

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Solution

Given: equation $5 x^{2} + (9 - 4 k) x + 2 k^{2} = 0$ and $5 x + 9 = 0$ have same value of $x$
To find the values of $k$
Sol: $5 x + 9 = 0 ⟹ x = - \frac{9}{5}$
Substituting this value of $x$ in first equation, we get
$5 {(- \frac{9}{5})}^{2} + (9 - 4 k) (- \frac{9}{5}) + 2 k^{2} = 0 ⟹ \frac{81}{5} - \frac{81}{5} + \frac{36}{5} k + 2 k^{2} = 0 ⟹ 10 k^{2} + 36 k = 0 ⟹ 5 k^{2} + 18 k = 0$
$⟹ k = 0$ or $k = - \frac{18}{5}$

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