$8+x \sqrt{12}=\frac{8 x}{\sqrt{3}}$
Answer (2)
Simplify the square root: 12 = 2 3 .
Substitute and get: 8 + 2 x 3 = 3 8 x .
Multiply by 3 : 8 3 + 6 x = 8 x .
Solve for x : x = 4 3 .
Explanation
Problem Setup We are given the equation 8 + x 12 = 3 8 x and we want to solve for x .
Simplifying the Square Root First, let's simplify the square root 12 . We can rewrite it as 12 = 4 × 3 = 4 × 3 = 2 3 . Substituting this into the original equation, we get 8 + x ( 2 3 ) = 3 8 x .
Eliminating the Fraction Now, we have the equation 8 + 2 x 3 = 3 8 x . To eliminate the fraction, we multiply both sides of the equation by 3 . This gives us 3 ( 8 + 2 x 3 ) = 3 ( 3 8 x ) , which simplifies to 8 3 + 2 x ( 3 ) 2 = 8 x .
Simplifying the Equation Since ( 3 ) 2 = 3 , the equation becomes 8 3 + 2 x ( 3 ) = 8 x , which simplifies to 8 3 + 6 x = 8 x .
Isolating x Next, we want to isolate x . Subtract 6 x from both sides of the equation to get 8 3 = 8 x − 6 x , which simplifies to 8 3 = 2 x .
Solving for x Finally, divide both sides by 2 to solve for x : 2 8 3 = x . This gives us x = 4 3 .
Examples
Imagine you are building a triangular garden bed and need to calculate the length of one side given a relationship involving square roots and another side. The equation we solved is similar to those that arise in geometric problems, where simplifying radicals and solving for unknowns are crucial for determining dimensions and ensuring accurate construction.
We solved the equation 8 + x 12 = 3 8 x by simplifying the square root to 2 3 , eliminating fractions, and isolating x . This results in x = 4 3 . The steps include simplifying, combining like terms, and solving for the variable.
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