Sure! Let's solve the equation [tex]\(\sqrt{3x + 4} = \sqrt{3x} + 4\)[/tex] step by step to determine which statement accurately describes the method needed.
1. Square both sides once:
Square both sides of the equation:
[tex]\[
\left(\sqrt{3x + 4}\right)^2 = \left(\sqrt{3x} + 4\right)^2
\][/tex]
Simplify:
[tex]\[
3x + 4 = (\sqrt{3x} + 4)^2
\][/tex]
2. Expand the right-hand side:
[tex]\[
3x + 4 = (\sqrt{3x})^2 + 2 \cdot \sqrt{3x} \cdot 4 + 4^2
\][/tex]
Simplify further:
[tex]\[
3x + 4 = 3x + 8\sqrt{3x} + 16
\][/tex]
3. Isolate the radical term:
Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[
4 = 8\sqrt{3x} + 16
\][/tex]
Subtract 16 from both sides:
[tex]\[
4 - 16 = 8\sqrt{3x}
\][/tex]
Simplify:
[tex]\[
-12 = 8\sqrt{3x}
\][/tex]
4. Solve for [tex]\(\sqrt{3x}\)[/tex]:
Divide both sides by 8:
[tex]\[
\frac{-12}{8} = \sqrt{3x}
\][/tex]
[tex]\[
-\frac{3}{2} = \sqrt{3x}
\][/tex]
5. Analyze the result:
[tex]\[
-\frac{3}{2} = \sqrt{3x}
\][/tex]
Since the square root of a number [tex]\(\sqrt{3x}\)[/tex] is always non-negative, but the left side of the equation is negative [tex]\(-\frac{3}{2}\)[/tex], we have a contradiction. This indicates that there are no real solutions to the equation.
Given this step-by-step analysis, it is evident that we squared both sides of the equation once, but eventually found that there is no real solution. Thus, the correct statement which describes the solving approach is:
Square both sides once and then solve the resulting equation.
