a) Explain the meaning of the term "principal square root."
b) Solve x^2 = 25.
c) Solve x = √25.
d) Compare and explain the solutions to parts b and c.
Answer (2)
Principal square root is the non-negative square root of a number.
Solving x 2 = 25 yields two solutions: x = 5 and x = − 5 .
Solving x = 25 yields one solution: x = 5 , the principal square root.
The key difference is that x 2 = 25 asks for all numbers that square to 25, while x = 25 asks for the principal square root.
The final answer is 5 .
Explanation
Problem Analysis Let's break down this problem step by step. We'll start by defining the principal square root, then solve the given equations, and finally compare the solutions.
Understanding Principal Square Root The principal square root of a non-negative number is its non-negative square root. For example, the principal square root of 25 is 5, not -5. When we see the square root symbol , it refers to the principal square root.
Solving x^2 = 25 To solve x 2 = 25 , we need to find all values of x that, when squared, equal 25. Taking the square root of both sides, we get x = ± 25 . This means x can be either the positive or negative square root of 25. Therefore, the solutions are x = 5 and x = − 5 .
Solving x = √25 To solve x = 25 , we are looking for the principal square root of 25. As we defined earlier, the principal square root is the non-negative square root. Therefore, x = 5 .
Comparing the Solutions The equation x 2 = 25 has two solutions: x = 5 and x = − 5 . This is because both 5 2 = 25 and ( − 5 ) 2 = 25 . The equation x = 25 has only one solution: x = 5 . This is because the square root symbol refers to the principal (non-negative) square root.
Examples
Understanding square roots is essential in many real-world applications. For instance, when calculating the length of the side of a square given its area, you need to find the square root. If a square has an area of 25 square meters, the length of each side is 25 = 5 meters. Similarly, in physics, square roots are used in formulas such as calculating the speed of an object or the distance it travels under constant acceleration. Knowing the difference between finding all solutions to x 2 = a and finding the principal square root x = a is crucial for accurate problem-solving.
The principal square root of a number is its non-negative square root. Solving x 2 = 25 gives two solutions: x = 5 and x = − 5 , while x = 25 gives the single solution x = 5 . Hence, the first equation asks for all square roots, while the second specifies the principal one.
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