Solve by factoring or finding square roots.
[tex](x+5)^2=36[/tex]
Answer (1)
Take the square root of both sides of the equation: x + 5 = ± 6 .
Solve for x when x + 5 = 6 : x = 6 − 5 = 1 .
Solve for x when x + 5 = − 6 : x = − 6 − 5 = − 11 .
The solutions are x = 1 and x = − 11 , so the final answer is 1 , − 11 .
Explanation
Understanding the Problem We are given the equation ( x + 5 ) 2 = 36 and asked to solve for x by factoring or finding square roots.
Taking the Square Root To solve this equation, we can take the square root of both sides. Remember that when we take the square root, we must consider both the positive and negative roots.
Simplifying the Equation Taking the square root of both sides, we get:
( x + 5 ) 2 = ± 36
x + 5 = ± 6
Solving for x Now we have two separate equations to solve:
x + 5 = 6
x + 5 = − 6
Solving the First Equation For the first equation, x + 5 = 6 , we subtract 5 from both sides to isolate x :
x = 6 − 5
x = 1
Solving the Second Equation For the second equation, x + 5 = − 6 , we subtract 5 from both sides to isolate x :
x = − 6 − 5
x = − 11
Final Answer Therefore, the solutions to the equation ( x + 5 ) 2 = 36 are x = 1 and x = − 11 .
Examples
Imagine you're designing a square garden with sides of length x + 5 meters, and you want the area of the garden to be 36 square meters. This problem helps you find the possible values of x , which represents how much longer each side is than 5 meters. Solving this equation allows you to determine the actual dimensions of the garden, ensuring it meets your desired area. This type of problem is also applicable in various engineering and construction scenarios where dimensions need to be precise.
