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Which statement describes how to solve [tex]\sqrt{3x + 4} = \sqrt{3x} + 4[/tex]?

A. Square both sides once and then solve the resulting linear equation.
B. Square both sides once and then solve the resulting quadratic equation.
C. Square both sides twice and then solve the resulting linear equation.
D. Square both sides twice and then solve the resulting quadratic equation.


Sagot :

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.