Problem 100 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The number that satisfies the given conditions is \(7\).

Step 1: Formulate the equation

Translate the word problem into an algebraic equation. So, \(\sqrt{x - 3} = x - 5\).

Step 2: Solve the square root equation

Square both sides of the equation to get rid of the square root. So, \((\sqrt{x - 3})^2 = (x - 5)^2\), which simplifies to \(x - 3 = x^2 - 10x + 25\). Standardize the formula to bring it to a quadratic equation form \(x^2 -11x + 28 = 0\).

Step 3: Find the roots

Solve the quadratic equation using the formula for roots \((x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a})\), where \(a = 1,\) \(b = -11,\) and \(c = 28.\) This yields two potential solutions for \(x:\) \(x = 7\) and \(x = 4\).

Step 4: Check the solutions

Substitute the potential solutions back into the original equation to verify. Substituting \(x = 7\) gives us \(\sqrt{7 - 3} = 7 - 5\) => \(\sqrt{4} = 2\), which is true. But substituting \(x = 4\) gives us \(\sqrt{4 - 3} = 4 - 5\) => \(\sqrt{1} = -1\), which is not correct. So, the only number that satisfies the given conditions is \(7\).