Question:
Determine. if 3 is a root of the equation given below:
$\sqrt{x^{2}-4 x+3}+\sqrt{x^{2}-9}=\sqrt{4 x^{2}-14 x+16}$
Solution:
We have been given that,
$\sqrt{x^{2}-4 x+3}+\sqrt{x^{2}-9}=\sqrt{4 x^{2}-14 x+16}$
We have to check whether x = 3 is the solution of the given equation or not.
Now, if $x=3$ is a root of the above quadratic equation, then it should satisfy the whole. So substituting $x=3$ in the above equation, we have,
Left hand side
$=\sqrt{(3)^{2}-4(3)+3}+\sqrt{(3)^{2}-9}$
$=\sqrt{0}+\sqrt{0}$
$=0$
Right hand side
$=\sqrt{4\left(3^{2}\right)-14(3)+16}$
$=\sqrt{36-42+16}$
$=\sqrt{10}$
Now since, we can see from above that left hand side and right hand side are not equal. Therefore $x=3$ is not the solution of the given quadratic equation.