Determine Whether the Given Values Are Solutions of the Equation
Question:
Determine whether the given values are solution of the given equation or not:
\[ x^2-3\sqrt3\,x+6=0 \]
(i) \(x=\sqrt3\)
(ii) \(x=-2\sqrt3\)
Solution
A value of \(x\) is a solution if it makes the left-hand side equal to zero.
Checking \(x=\sqrt3\)
Substitute \(x=\sqrt3\):
\[ (\sqrt3)^2-3\sqrt3(\sqrt3)+6 \]
\[ =3-9+6 \]
\[ =0 \]
Therefore, \(x=\sqrt3\) is a solution of the equation.
Checking \(x=-2\sqrt3\)
Substitute \(x=-2\sqrt3\):
\[ (-2\sqrt3)^2-3\sqrt3(-2\sqrt3)+6 \]
\[ =12+18+6 \]
\[ =36 \]
Since \(36\ne0\), the equation is not satisfied.
Therefore, \(x=-2\sqrt3\) is not a solution of the equation.
Answer
\[ \boxed{x=\sqrt3 \text{ is a solution and } x=-2\sqrt3 \text{ is not a solution}} \]