What is the solution to the equation x² = 16? Explain.
Answer (2)
Recognize that the equation is x 2 = 16 .
Take the square root of both sides: x 2 = 16 .
Identify both positive and negative roots: x = ± 4 .
State the solutions: x = 4 and x = − 4 , so the answer is x = ± 4 .
Explanation
Understanding the Problem We are asked to find the solution to the equation x 2 = 16 . This means we need to find all values of x that, when squared, equal 16. Remember that squaring a number means multiplying it by itself.
Taking the Square Root To solve this, we take the square root of both sides of the equation. The square root of a number is a value that, when multiplied by itself, gives the original number. So, we have x 2 = 16 .
Considering Both Positive and Negative Roots We know that 16 = 4 because 4 × 4 = 16 . However, we must also remember that negative numbers, when squared, become positive. So, ( − 4 ) × ( − 4 ) = 16 as well. Therefore, 16 can be both 4 and -4.
Finding the Solutions Thus, the solutions to the equation x 2 = 16 are x = 4 and x = − 4 .
Examples
Understanding square roots is essential in many real-world applications. For example, if you're designing a square garden with an area of 16 square meters, you need to find the length of each side. Since the area of a square is side * side, you would solve the equation s 2 = 16 to find that each side should be 4 meters long. This concept is also used in construction, engineering, and even art to ensure accurate measurements and proportions.
The solutions to the equation x 2 = 16 are x = 4 and x = − 4 . This is determined by taking the square root of both sides of the equation. Therefore, the values of x that satisfy the equation are four and negative four.
;
