Determining the complete set of real square roots of 100 is a fundamental exercise that reveals the underlying symmetry of quadratic equations. While the result may seem straightforward, a thorough analysis distinguishes between the principal root and the full solution set. This exploration requires understanding the definition of a square root as a value that, when multiplied by itself, yields the original number.
Defining the Square Root Operation
The square root of a number represents a value that produces the original number when squared. For the integer 100, we are searching for all real numbers x such that x² = 100 . This equation forms the basis of the calculation, and solving it requires considering both the positive and negative possibilities, as squaring either a positive or negative number results in a positive product.
The Principal Square Root
In mathematical notation, the radical symbol √ specifically denotes the principal square root, which is always the non-negative root. Applying this to 100, the principal square root is written as √100 and equals 10. This value is the standard output of the square root function and is the root most frequently encountered in geometric and scientific contexts where a physical length or magnitude is required.
Calculating the Positive Root
To find the positive root, one looks for the number that multiplies by itself to reach 100. Factoring 100 into its prime components as 2 × 2 × 5 × 5 allows the expression to be grouped as (2 × 5) × (2 × 5) , which is 10 × 10 . Therefore, the principal root is definitively 10, satisfying the condition where the input and output of the squaring function align positively.
The Complete Set of Real Roots
However, the question asks for all real square roots, which requires expanding the view beyond the principal value. Since the problem involves solving the equation x² = 100 , we must account for the fact that multiplying two negative numbers also yields a positive result. This introduces a second solution that is the additive inverse of the first.
Verification of Negative Roots
To verify the second solution, consider the number -10. When squared, the expression becomes (-10) × (-10) , which equals 100 according to the rule that a negative times a negative is a positive. Consequently, -10 is also a valid real square root of 100, demonstrating that the solution set includes both opposite values.
The Solution Set
Summarizing the analysis, the equation x² = 100 has two distinct real solutions. The complete set of real square roots of 100 consists of the positive number 10 and the negative number -10. No other real numbers satisfy the condition, as fractions or other integers would either square to a value too small or too large to equal 100.
