Select whether the equation has a solution or not.
[tex]\sqrt{4 x}=2 \sqrt{2 x-3}[/tex]
A. no roots
B. roots
Answer (2)
Square both sides of the equation: ( 4 x ) 2 = ( 2 2 x − 3 ) 2 which simplifies to 4 x = 4 ( 2 x − 3 ) .
Simplify and solve for x : 4 x = 8 x − 12 leads to x = 3 .
Check the solution by substituting x = 3 back into the original equation: 4 ( 3 ) = 2 2 ( 3 ) − 3 .
Since 12 = 2 3 , the solution x = 3 is valid, and the equation has a solution. $\boxed{roots}
Explanation
Problem Analysis We are given the equation 4 x = 2 2 x − 3 and asked to determine if it has a solution.
Squaring Both Sides To solve this, we first square both sides of the equation to eliminate the square roots: ( 4 x ) 2 = ( 2 2 x − 3 ) 2
Simplifying the Equation Simplifying the equation gives us: 4 x = 4 ( 2 x − 3 ) 4 x = 8 x − 12
Solving for x Now, we solve for x : 4 x − 8 x = − 12 − 4 x = − 12 x = − 4 − 12 x = 3
Checking the Solution We need to check if this solution is valid by substituting x = 3 back into the original equation: 4 ( 3 ) = 2 2 ( 3 ) − 3 12 = 2 6 − 3 12 = 2 3
Conclusion Since 12 = 4 ⋅ 3 = 2 3 , the solution x = 3 is valid. Therefore, the equation has a solution.
Examples
Consider a scenario where you need to determine the length of a side of a square garden such that its area is related to the area of a rectangular garden in a specific way. Solving equations with square roots, like the one above, can help you find the dimensions that satisfy the given relationship between the gardens' areas. This type of problem arises in various fields, including engineering, physics, and computer graphics, where relationships between quantities are expressed using equations involving square roots.
The equation 4 x = 2 2 x − 3 has a solution, specifically x = 3 . After checking this value, it satisfies the original equation. Thus, I would select option B: roots.
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