To determine if [tex]\( x = 3 \)[/tex] is a root of the equation
[tex]\[ \sqrt{x^2-4x+3} + \sqrt{x^2-9} = \sqrt{4x^2-14x+16}, \][/tex]
we need to evaluate both sides of the equation separately at [tex]\( x = 3 \)[/tex] and then compare the results.
First, we calculate the left side of the equation:
[tex]\[ \sqrt{x^2-4x+3} + \sqrt{x^2-9}. \][/tex]
Substituting [tex]\( x = 3 \)[/tex] into each term, we get:
[tex]\[
\sqrt{(3)^2 - 4(3) + 3} = \sqrt{9 - 12 + 3} = \sqrt{0} = 0
\][/tex]
and
[tex]\[
\sqrt{(3)^2 - 9} = \sqrt{9 - 9} = \sqrt{0} = 0.
\][/tex]
So the left side becomes:
[tex]\[
0 + 0 = 0.
\][/tex]
Next, we calculate the right side of the equation:
[tex]\[ \sqrt{4x^2 - 14x + 16}. \][/tex]
Substituting [tex]\( x = 3 \)[/tex], we get:
[tex]\[
\sqrt{4(3)^2 - 14(3) + 16} = \sqrt{36 - 42 + 16} = \sqrt{10} \approx 3.162.
\][/tex]
Thus, the right side is approximately [tex]\( 3.162 \)[/tex].
Comparing both sides of the equation for [tex]\( x = 3 \)[/tex]:
[tex]\[
\sqrt{3^2 - 4(3) + 3} + \sqrt{3^2 - 9} = 0 \quad \text{and} \quad \sqrt{4(3)^2 - 14(3) + 16} \approx 3.162.
\][/tex]
Clearly, the left side [tex]\( 0 \neq 3.162 \)[/tex] from the right side. Therefore, [tex]\( x = 3 \)[/tex] does not satisfy the equation, meaning that [tex]\( x = 3 \)[/tex] is not a root of the equation
[tex]\[
\sqrt{x^2-4x+3} + \sqrt{x^2-9} = \sqrt{4x^2-14x+16}.
\][/tex]
