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

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Let the number be represented by the variable \(x\).

Translate the problem statement into an equation: "If a number is decreased by 3" becomes \(x - 3\), and "the principal square root of this difference is 5 less than the number" becomes \(\sqrt{x - 3} = x - 5\).

Square both sides of the equation to eliminate the square root: \((\sqrt{x - 3})^2 = (x - 5)^2\), which simplifies to \(x - 3 = (x - 5)^2\).

Expand the right side: \((x - 5)^2 = x^2 - 10x + 25\), so the equation becomes \(x - 3 = x^2 - 10x + 25\).

Rearrange the equation to standard quadratic form by moving all terms to one side: \(0 = x^2 - 10x + 25 - x + 3\), which simplifies to \(0 = x^2 - 11x + 28\). This quadratic equation can now be solved for \(x\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Setting up Algebraic Equations

This involves translating a word problem into an algebraic equation using variables to represent unknown quantities. Identifying relationships described in the problem allows you to form equations that can be solved systematically.

Square Roots and Principal Square Root

The principal square root of a number is the non-negative root. Understanding how to work with square roots, including isolating the root and squaring both sides of an equation, is essential for solving equations involving roots.

Solving Quadratic Equations

After forming an equation, you may need to rearrange it into a quadratic form and solve using factoring, completing the square, or the quadratic formula. Checking solutions is important because squaring can introduce extraneous roots.

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