If a number is decreased by 3, the principal square root of this difference is 5 less than the number. Find the number(s).

Verified step by step guidance

1. Let the number be represented by the variable \( x \).
2. According to the problem, if the number is decreased by 3, we have the expression \( x - 3 \).
3. The principal square root of this difference is given by \( \sqrt{x - 3} \).
4. This square root is 5 less than the number, which can be expressed as \( x - 5 \).
5. Set up the equation: \( \sqrt{x - 3} = x - 5 \) and solve for \( x \) by squaring both sides and simplifying.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators. In this problem, we need to translate the verbal statement into an algebraic equation. Understanding how to form expressions from words is crucial for setting up the problem correctly.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. In this question, we are dealing with the principal square root, which is the non-negative root. Recognizing how to manipulate square roots in equations is essential for solving the problem.

Solving Equations

Solving equations involves finding the value(s) of the variable that make the equation true. This process often includes isolating the variable, applying inverse operations, and checking solutions. Mastery of solving equations is necessary to find the number(s) in this problem.