Determine if 3 is a Root of the Equation
Question:
Determine if \(3\) is a root of the equation:
\[ \sqrt{x^2-4x+3}+\sqrt{x^2-9} = \sqrt{4x^2-14x+16} \]
Solution
To check whether \(3\) is a root, substitute \(x=3\) into the equation.
Left-Hand Side (LHS)
\[ \sqrt{3^2-4(3)+3}+\sqrt{3^2-9} \]
\[ =\sqrt{9-12+3}+\sqrt{9-9} \]
\[ =\sqrt{0}+\sqrt{0} \]
\[ =0+0 \]
\[ =0 \]
Right-Hand Side (RHS)
\[ \sqrt{4(3)^2-14(3)+16} \]
\[ =\sqrt{36-42+16} \]
\[ =\sqrt{10} \]
Comparison
\[ \text{LHS}=0 \]
\[ \text{RHS}=\sqrt{10} \]
Since \[ 0 \ne \sqrt{10}, \] the equation is not satisfied.
Answer
Therefore, \(3\) is not a root of the given equation.
\[ \boxed{3 \text{ is not a root of the equation}} \]