Solve the following equation by the square root property.
[tex](3 x-2)^2=29[/tex]
The solution set is { }. (Simplify your answer. Type an exact answer, using radicals and [tex]i[/tex] as needed. Use a comma to separate answers as needed.)
Answer (2)
Apply the square root property to the equation ( 3 x − 2 ) 2 = 29 , resulting in 3 x − 2 = ± 29 .
Add 2 to both sides: 3 x = 2 ± 29 .
Divide by 3 to solve for x : x = 3 2 ± 29 .
The solution set is { 3 2 + 29 , 3 2 − 29 } .
Explanation
Understanding the Problem We are given the equation ( 3 x − 2 ) 2 = 29 and asked to solve it using the square root property. This means we need to take the square root of both sides of the equation to isolate the term with x .
Applying the Square Root Property Taking the square root of both sides, we get: ( 3 x − 2 ) 2 = ± 29 This simplifies to: 3 x − 2 = ± 29
Isolating the Term with x Now, we want to isolate x . First, add 2 to both sides of the equation: 3 x = 2 ± 29
Solving for x Next, divide both sides by 3: x = 3 2 ± 29
Final Solutions Therefore, the two solutions are: x = 3 2 + 29 and x = 3 2 − 29 So, the solution set is { 3 2 + 29 , 3 2 − 29 } .
Final Answer Thus, the solution set is { 3 2 + 29 , 3 2 − 29 } .
Examples
The square root property is useful in physics when dealing with equations involving squares of variables, such as in energy calculations or kinematic equations. For example, if you have an equation relating the square of an object's velocity to its kinetic energy, you can use the square root property to solve for the velocity. Suppose the kinetic energy K E of an object is given by K E = 2 1 m v 2 , where m is the mass and v is the velocity. If you know the kinetic energy and mass, you can solve for v using the square root property.
By applying the square root property to the equation ( 3 x − 2 ) 2 = 29 , we isolate x and find the solutions to be 3 2 + 29 and 3 2 − 29 . The complete solution set is { 3 2 + 29 , 3 2 − 29 } .
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