Determine Whether the Given Values Are Solutions of the Equation
Question:
Determine whether the given values are solution of the given equation or not:
\[ x^2-\sqrt2\,x-4=0 \]
(i) \(x=-\sqrt2\)
(ii) \(x=-2\sqrt2\)
Solution
A value of \(x\) is a solution if it makes the left-hand side equal to zero.
Checking \(x=-\sqrt2\)
Substitute \(x=-\sqrt2\):
\[ (-\sqrt2)^2-\sqrt2(-\sqrt2)-4 \]
\[ =2+2-4 \]
\[ =0 \]
Therefore, \(x=-\sqrt2\) is a solution of the equation.
Checking \(x=-2\sqrt2\)
Substitute \(x=-2\sqrt2\):
\[ (-2\sqrt2)^2-\sqrt2(-2\sqrt2)-4 \]
\[ =8+4-4 \]
\[ =8 \]
Since \(8\ne0\), the equation is not satisfied.
Therefore, \(x=-2\sqrt2\) is not a solution of the equation.
Answer
\[ \boxed{x=-\sqrt2 \text{ is a solution and } x=-2\sqrt2 \text{ is not a solution}} \]