$(7 x+2)^2=16$
The solution set is { }. (Simplify your answers. Type exact answers, using radicals as needed. Use a comma to separate answers as needed.)
Answer (2)
Take the square root of both sides of the equation: 7 x + 2 = ± 4 .
Solve for x in the first case: 7 x + 2 = 4 , which gives x = 7 2 .
Solve for x in the second case: 7 x + 2 = − 4 , which gives x = − 7 6 .
The solution set is { 7 2 , − 7 6 } .
Explanation
Understanding the Problem We are given the equation ( 7 x + 2 ) 2 = 16 . Our goal is to find the solution set for x , simplifying the answers and using radicals as needed. If there are multiple solutions, we will separate them by commas.
Taking the Square Root To solve the equation, we first take the square root of both sides: ( 7 x + 2 ) 2 = 16
This gives us: 7 x + 2 = ± 4
Solving for x Now we have two cases to consider:
Case 1: 7 x + 2 = 4 Subtract 2 from both sides: 7 x = 4 − 2
7 x = 2 Divide by 7: x = 7 2
Case 2: 7 x + 2 = − 4 Subtract 2 from both sides: 7 x = − 4 − 2 7 x = − 6 Divide by 7: x = − 7 6
Final Answer Therefore, the solution set is { 7 2 , − 7 6 } .
Examples
Imagine you are designing a square garden and want to know the possible lengths of the sides if the area of a smaller square section within the garden is defined by the equation ( 7 x + 2 ) 2 = 16 . Solving this equation gives you the possible values for x , which could represent a scaled dimension related to the garden's design. Understanding how to solve such equations allows you to plan and adjust the dimensions of your garden effectively, ensuring it meets your design specifications. This type of problem also appears in physics, where you might be calculating the position of an object based on its squared displacement.
To solve ( 7 x + 2 ) 2 = 16 , we take the square root of both sides to get 7 x + 2 = ± 4 . Solving the resulting equations gives us the solution set { \frac{2}{7}, -\frac{6}{7} }.
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