To solve the equation [tex]\(\sqrt{1-3x} = x + 3\)[/tex], we'll follow a step-by-step approach to verify the given possible solutions: [tex]\(x = -8\)[/tex] and [tex]\(x = -1\)[/tex].
1. Test [tex]\( x = -8 \)[/tex] in the original equation:
[tex]\[
\sqrt{1-3(-8)} = -8 + 3
\][/tex]
Simplify inside the square root:
[tex]\[
\sqrt{1 + 24} = -5
\][/tex]
This reduces to:
[tex]\[
\sqrt{25} = -5
\][/tex]
which simplifies to:
[tex]\[
5 \neq -5
\][/tex]
The solution [tex]\( x = -8 \)[/tex] does not satisfy the original equation.
2. Test [tex]\( x = -1 \)[/tex] in the original equation:
[tex]\[
\sqrt{1-3(-1)} = -1 + 3
\][/tex]
Simplify inside the square root:
[tex]\[
\sqrt{1 + 3} = 2
\][/tex]
This reduces to:
[tex]\[
2 = 2
\][/tex]
The solution [tex]\( x = -1 \)[/tex] satisfies the original equation.
Therefore, among the given solutions [tex]\( x = -8 \)[/tex] and [tex]\( x = -1 \)[/tex], the only correct solution is [tex]\( x = -1 \)[/tex].
So the solution of [tex]\(\sqrt{1-3x} = x+3\)[/tex] is:
[tex]\[ \boxed{x = -1} \][/tex]
