Solve $(y+6)^2=4$ where $y$ is a real number. Simplify your answer as much as possible. (If there is more than one solution, separate them with commas.) $y=$
Answer (2)
Take the square root of both sides of the equation: y + 6 = ± 2 .
Solve for y in the two equations: y + 6 = 2 and y + 6 = − 2 .
For y + 6 = 2 , subtract 6 from both sides to get y = − 4 .
For y + 6 = − 2 , subtract 6 from both sides to get y = − 8 . The solutions are − 4 , − 8 .
Explanation
Problem Analysis We are given the equation ( y + 6 ) 2 = 4 and we want to solve for y , where y is a real number.
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 need to consider both the positive and negative square roots. So, we have y + 6 = ± 4 , which simplifies to y + 6 = ± 2 .
Two Separate Equations Now we have two separate equations to solve:
y + 6 = 2
y + 6 = − 2
Solving the First Equation For the first equation, y + 6 = 2 , we subtract 6 from both sides to isolate y :
y = 2 − 6 = − 4
Solving the Second Equation For the second equation, y + 6 = − 2 , we subtract 6 from both sides to isolate y :
y = − 2 − 6 = − 8
Final Answer Therefore, the solutions are y = − 4 and y = − 8 .
Examples
Imagine you're designing a square garden with sides of length y + 6 meters, and you want the area of the garden to be exactly 4 square meters. This problem helps you find the possible values for y , which represents how much you need to adjust the side length from a base of 6 meters to achieve the desired area. Understanding how to solve such equations is crucial in various fields like engineering, where precise measurements are essential for designing structures and systems.
The solutions to the equation ( y + 6 ) 2 = 4 are y = − 4 and y = − 8 . The steps involved taking the square root of both sides, setting up two equations, and solving for y . Finally, we identified both solutions for y .
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