Question

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

Answer 1

Hint: In this question we are given a quadratic equation with an unknown number $k$. Moreover we are given a root as well. Since we know the quadratic equation has two roots so we have to find the other root also. But for that we would need to find the value of the unknown $k$. So, to do this we would simply use the property that the root is going to satisfy the equation and then plug in the value of the root to find the value of $k$. After $k$ has been found, we will solve the quadratic equation and find the other root as well.

Complete step by step answer:
We have $x=1$ as the roots of the given quadratic equation $x^2+kx+3=0$. We know that the roots of a quadratic equation satisfy the equation. So, we simply put $x=1$ in the quadratic equation.
By putting 1 in place of $x$, we get:
$1^2+k\times 1+3=0$
$\implies 1+k+3=0$
$\implies k=-4$
Putting this value of $k$, we obtain the quadratic equation,
$x^2-4x+3=0$
We can dissolve this equation as follows:
$x^2-3x-x+3=0$
$\implies x\left(x-3\right)-\left(x-3\right)=0$
Taking $\left(x-3\right)$ common, we get:
 $\implies \left(x-3\right)\left(x-1\right)=0$
Hence, the roots obtained are $x=1$ and $x=3$. Therefore, $x=3$ is the other root.

Note: While dissolving the quadratic equation, make sure that you find two numbers whose product is the constant term and sum is the coefficient of linear term in $x$. Moreover, since one root was already given, you could have also performed long division by dividing the given quadratic equation by $x-1$.