Solve $(x-3)^2=49$. Select the values of $x$.
Answer (1)
Take the square root of both sides of the equation: x − 3 = ± 7 .
Solve for x in the first case: x − 3 = 7 , which gives x = 10 .
Solve for x in the second case: x − 3 = − 7 , which gives x = − 4 .
The solutions are x = 10 and x = − 4 , so the final answer is − 4 , 10 .
Explanation
Understanding the Problem We are given the equation ( x − 3 ) 2 = 49 and asked to solve for x . This means we need to find the values of x that make the equation true.
Taking the Square Root To solve the equation, we can take the square root of both sides. Remember that when we take the square root, we need to consider both the positive and negative roots. So, we have ( x − 3 ) 2 = ± 49 . This simplifies to x − 3 = ± 7 .
Splitting into Two Equations Now we have two separate equations to solve:
x − 3 = 7
x − 3 = − 7
Solving the First Equation Solving the first equation, x − 3 = 7 , we add 3 to both sides to isolate x : x = 7 + 3 = 10
Solving the Second Equation Solving the second equation, x − 3 = − 7 , we add 3 to both sides to isolate x : x = − 7 + 3 = − 4
Final Answer Therefore, the solutions to the equation ( x − 3 ) 2 = 49 are x = 10 and x = − 4 .
Examples
Imagine you are designing a square garden with an area of 49 square feet. You want to place it in a rectangular yard such that the garden's side is 'x-3' feet away from the edge of the yard. The equation (x-3)^2 = 49 helps you determine the possible values for x, which represent the total length of one side of the yard. By solving this equation, you find the possible lengths that satisfy the garden's area requirement within the yard.
