Solve the equation.
[tex]\begin{array}{c}
\sqrt{3 x-4}=2 \sqrt{x-5} \\
x=[?]\end{array}[/tex]
Answer (1)
Square both sides of the equation: 3 x − 4 = 4 ( x − 5 ) .
Expand and simplify: 3 x − 4 = 4 x − 20 .
Isolate x: x = 16 .
Verify the solution in the original equation: 3 ( 16 ) − 4 = 2 16 − 5 , which simplifies to 2 11 = 2 11 .
The solution is x = 16 .
Explanation
Problem Analysis We are given the equation 3 x − 4 = 2 x − 5 and we need to solve for x .
Squaring Both Sides To eliminate the square roots, we square both sides of the equation: ( 3 x − 4 ) 2 = ( 2 x − 5 ) 2
Simplifying Simplifying the equation, we get: 3 x − 4 = 4 ( x − 5 )
Expanding Expanding the right side of the equation, we have: 3 x − 4 = 4 x − 20
Isolating x Now, we isolate x by subtracting 3 x from both sides: 3 x − 4 − 3 x = 4 x − 20 − 3 x − 4 = x − 20
Solving for x Adding 20 to both sides to solve for x , we get: x = 16
Checking the Solution Now, we need to check if the solution is valid by substituting x = 16 into the original equation: 3 ( 16 ) − 4 = 2 16 − 5 48 − 4 = 2 11 44 = 2 11 2 11 = 2 11 The solution is valid.
Checking Non-Negativity Conditions Also, we need to check the non-negativity conditions: 3 x − 4 ≥ 0 and x − 5 ≥ 0 .
For x = 16 , we have: 3 ( 16 ) − 4 = 44 ≥ 0 and 16 − 5 = 11 ≥ 0 . Both conditions are satisfied.
Examples
Imagine you are designing a bridge and need to calculate the length of support cables. The equation 3 x − 4 = 2 x − 5 is similar to equations used to model the tension and length of cables under certain loads. Solving such equations ensures the bridge is stable and safe. Understanding how to manipulate and solve equations with square roots is crucial in engineering for accurate and reliable structural designs.
