Solve for x: \(\sqrt{5x+66} + 2 = 8\)
Understand the Problem
The problem is asking to solve the algebraic equation (\sqrt{5x+66} + 2 = 8) for the variable x. This will involve isolating the square root term, squaring both sides, and then solving the resulting linear equation.
Answer
$x = -6$
Answer for screen readers
$x = -6$
Steps to Solve
- Isolate the square root term
Subtract 2 from both sides of the equation to isolate the square root: $$ \sqrt{5x+66} + 2 - 2 = 8 - 2 $$ $$ \sqrt{5x+66} = 6 $$
- Square both sides of the equation
To eliminate the square root, square both sides of the equation: $$ (\sqrt{5x+66})^2 = 6^2 $$ $$ 5x + 66 = 36 $$
- Isolate the variable term
Subtract 66 from both sides of the equation to isolate the term with $x$: $$ 5x + 66 - 66 = 36 - 66 $$ $$ 5x = -30 $$
- Solve for x
Divide both sides of the equation by 5 to solve for $x$: $$ \frac{5x}{5} = \frac{-30}{5} $$ $$ x = -6 $$
$x = -6$
More Information
We can check the solution by substituting $x = -6$ into the original equation: $$ \sqrt{5(-6)+66} + 2 = \sqrt{-30+66} + 2 = \sqrt{36} + 2 = 6 + 2 = 8 $$ The solution checks out.
Tips
A common mistake is forgetting to isolate the square root before squaring both sides. If you don't isolate the square root first, you will have a more complicated expression to square, and you are more likely to make an error. Another common mistake is making an arithmetic error when simplifying the equation.
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