What are the solutions of the quadratic equation $(x+3)^2=49$?
A. $x=-2$ and $x=-16$
B. $x=2$ and $x=\sqrt{-10}$
C. $x=4$ and $x=-10$
D. $x=40$ and $x=-58$
Answer (1)
Take the square root of both sides of the equation: x + 3 = ± 7 .
Solve for x when x + 3 = 7 : x = 7 − 3 = 4 .
Solve for x when x + 3 = − 7 : x = − 7 − 3 = − 10 .
The solutions are x = 4 and x = − 10 .
Explanation
Understanding the Problem We are given the quadratic equation ( x + 3 ) 2 = 49 and asked to find its solutions.
Taking the Square Root To solve this equation, we can take the square root of both sides. Remember that when taking the square root, we must consider both the positive and negative roots. So, we have x + 3 = ± 7 .
Splitting into Two Equations Now we have two separate equations to solve:
x + 3 = 7
x + 3 = − 7
Solving for x (Case 1) Solving the first equation, x + 3 = 7 , we subtract 3 from both sides to get x = 7 − 3 = 4 .
Solving for x (Case 2) Solving the second equation, x + 3 = − 7 , we subtract 3 from both sides to get x = − 7 − 3 = − 10 .
Final Answer Therefore, the solutions to the quadratic equation are x = 4 and x = − 10 .
Examples
Quadratic equations like this one appear in many real-world situations, such as calculating the trajectory of a ball, determining the dimensions of a garden, or modeling the growth of a population. For example, imagine you are designing a rectangular garden with an area of 49 square meters. One side is defined by (x+3). By solving the quadratic equation, you can find the possible values for x, which then helps you determine the actual dimensions of the garden. Understanding how to solve these equations allows you to tackle various practical problems.
