Answer :
To solve the equation \(\sqrt{3x + 4} = \sqrt{3x} + 4\), you need to proceed with the following detailed steps:
1. Square both sides of the equation:
[tex]\[ \left(\sqrt{3x + 4}\right)^2 = \left(\sqrt{3x} + 4\right)^2 \][/tex]
This results in:
[tex]\[ 3x + 4 = 3x + 8\sqrt{3x} + 16 \][/tex]
2. Simplify the equation:
Subtract \(3x\) from both sides:
[tex]\[ 4 = 8\sqrt{3x} + 16 \][/tex]
3. Isolate the square root term:
Subtract 16 from both sides:
[tex]\[ -12 = 8\sqrt{3x} \][/tex]
4. Solve for \(\sqrt{3x}\):
Divide both sides by 8:
[tex]\[ -1.5 = \sqrt{3x} \][/tex]
5. Square both sides again:
[tex]\[ (-1.5)^2 = (\sqrt{3x})^2 \][/tex]
This results in:
[tex]\[ 2.25 = 3x \][/tex]
6. Solve for \(x\):
Divide by 3:
[tex]\[ x = 0.75 \][/tex]
Based on this detailed procedure, the correct statement is:
Square both sides twice and then solve the resulting linear equation.
1. Square both sides of the equation:
[tex]\[ \left(\sqrt{3x + 4}\right)^2 = \left(\sqrt{3x} + 4\right)^2 \][/tex]
This results in:
[tex]\[ 3x + 4 = 3x + 8\sqrt{3x} + 16 \][/tex]
2. Simplify the equation:
Subtract \(3x\) from both sides:
[tex]\[ 4 = 8\sqrt{3x} + 16 \][/tex]
3. Isolate the square root term:
Subtract 16 from both sides:
[tex]\[ -12 = 8\sqrt{3x} \][/tex]
4. Solve for \(\sqrt{3x}\):
Divide both sides by 8:
[tex]\[ -1.5 = \sqrt{3x} \][/tex]
5. Square both sides again:
[tex]\[ (-1.5)^2 = (\sqrt{3x})^2 \][/tex]
This results in:
[tex]\[ 2.25 = 3x \][/tex]
6. Solve for \(x\):
Divide by 3:
[tex]\[ x = 0.75 \][/tex]
Based on this detailed procedure, the correct statement is:
Square both sides twice and then solve the resulting linear equation.
