If $x^{2} - 7 x + 10 = 0$ and $x^{2} - 5 x + c = 0$ have a common root, then find the value of $c$

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Solution

Let $x = x_{1}$ be the common roots
then,
$x_{1}^{2} - 7 x_{1} + 10 = 0$ ---- (i)
and $x_{1}^{2} - 5 x_{1} + c = 0$ ---- (ii)
subtracting (ii) from (i)
$- 7 x_{1} + 5 x_{1} + 10 - c = 0$
$c = - 2 x_{1} + 10$ ----- (iii)
on putting the value of $c$ in (ii)
$x_{1}^{2} - 7 x_{1} + 10 = 0$
$x_{1}^{2} - 2 x_{1} - 5 x_{1} + 10 = 0$
$x_{1} (x_{1} - 2) - 5 (x_{1} - 2) = 0$
$(x_{1} - 2) (x_{1} - 5) = 0$
$x_{1} = 2, 5$
on putting $x_{1} = 2, 5$ in equaton (iii), we get
$c = 6$ or $c = 0$

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