Find the value of $k$ for which $x = 1$ is a root of the equation $x^{2} + k x + 3 = 0$ . Also, find the other root.

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Solution

$x^{2} + k x + 3 = 0$ ...................(i)
Since $x = 1$ is the root therefore, $x = 1$ satisfies the eqn.(i)
$(1)^{2} + k (1) + 3 = 0$
$k + 4 = 0$
$k = - 4$
Therefore, $x^{2} + (- 4) x + 3 = x^{2} - 4 x + 3$
Hence, the roots are $x^{2} - 3 x - x + 3 = 0$
$x (x - 3) - 1 (x - 3) = 0$
$(x - 1) (x - 3) = 0$
Therefore, $x = 1, 3$

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Find the value of k for which x=1 is a root of the equation x2+kx+3=0. Also, find the other root.

x2+kx+3=0...................(i)Since x=1 is the root therefore, x=1 satisfies the eqn.(i)(1)2+k(1)+3=0k+4=0k=−4Therefore, x2+(−4)x+3=x2−4x+3Hence, the roots are x2−3x−x+3=0x(x−3)−1(x−3)=0(x−1)(x−3)=0Therefore, x=1,3

Find the value of $k$ for which $x = 1$ is a root of the equation $x^{2} + k x + 3 = 0$ . Also, find the other root.