Solve for x: √(6x + 48) - 1 = 5

Understand the Problem

The problem requires solving for 'x' in the given equation which involves a square root. The goal is to isolate x by performing algebraic operations such as adding constants, squaring both sides of the equation, and then dividing to get x by itself.

Answer

$x = 8$
Answer for screen readers

$x = 8$

Steps to Solve

  1. Isolate the square root term To begin, isolate the square root term by adding 5 to both sides of the equation: $$ \sqrt{3x+1} - 5 = 0 $$ $$ \sqrt{3x+1} = 5 $$

  2. Square both sides of the equation To eliminate the square root, square both sides of the equation: $$ (\sqrt{3x+1})^2 = 5^2 $$ $$ 3x+1 = 25 $$

  3. Isolate the x term Subtract 1 from both sides of the equation to isolate the term with x: $$ 3x + 1 - 1 = 25 - 1 $$ $$ 3x = 24 $$

  4. Solve for x Divide both sides of the equation by 3 to solve for x: $$ \frac{3x}{3} = \frac{24}{3} $$ $$ x = 8 $$

$x = 8$

More Information

The solution to the equation $\sqrt{3x+1} - 5 = 0$ is $x = 8$. We can check the solution by substituting x back into the original equation: $$ \sqrt{3(8)+1} - 5 = \sqrt{24+1} - 5 = \sqrt{25} - 5 = 5 - 5 = 0 $$ Since the equation holds true, $x = 8$ is indeed the solution.

Tips

A common mistake is forgetting to square the entire side of the equation when eliminating the square root. For example, if the equation were $\sqrt{a} + b = c$, you must write $(\sqrt{a} + b)^2 = c^2$, which expands to $a + 2b\sqrt{a} + b^2 = c^2$, and not just $a + b^2 = c^2$. Another mistake is not isolating the square root before squaring.

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