Determine Whether the Given Values Are Solutions of the Equation
Question:
Determine whether the given values are solution of the given equation or not:
\[ a^2x^2-3abx+2b^2=0 \]
(i) \(x=\frac{a}{b}\)
(ii) \(x=\frac{b}{a}\)
Solution
A value of \(x\) is a solution if it makes the left-hand side equal to zero.
Checking \(x=\frac{a}{b}\)
Substitute \(x=\frac{a}{b}\):
\[ a^2\left(\frac{a}{b}\right)^2-3ab\left(\frac{a}{b}\right)+2b^2 \]
\[ =\frac{a^4}{b^2}-3a^2+2b^2 \]
This expression is not identically equal to zero.
Hence, \[ x=\frac{a}{b} \] is not a solution in general.
Checking \(x=\frac{b}{a}\)
Substitute \(x=\frac{b}{a}\):
\[ a^2\left(\frac{b}{a}\right)^2-3ab\left(\frac{b}{a}\right)+2b^2 \]
\[ =b^2-3b^2+2b^2 \]
\[ =0 \]
Therefore, \[ x=\frac{b}{a} \] is a solution of the equation.
Answer
\[ \boxed{x=\frac{a}{b}\text{ is not a solution, whereas }x=\frac{b}{a}\text{ is a solution}} \]
Alternatively, factorizing:
\[ a^2x^2-3abx+2b^2 =(ax-b)(ax-2b) \]
The solutions are \[ x=\frac{b}{a} \quad\text{and}\quad x=\frac{2b}{a}. \]
Hence only \[ \boxed{x=\frac{b}{a}} \] among the given values satisfies the equation.