Find a natural number whose square diminished by \(84\) is equal to thrice of \(8\) more than the given number.
12
Step 1: Suppose the required natural number is \(x\).
Step 2: The question says "the square diminished by 84". That means: \(x^2 - 84\).
Step 3: It also says "is equal to thrice of (8 more than the number)". That means: \(3(x + 8)\).
Step 4: Write the equation: \(x^2 - 84 = 3(x + 8)\).
Step 5: Expand the right-hand side: \(x^2 - 84 = 3x + 24\).
Step 6: Bring everything to one side: \(x^2 - 3x - 108 = 0\).
Step 7: Factorise this quadratic equation: \(x^2 - 3x - 108 = (x - 12)(x + 9)\).
Step 8: From this, either \(x - 12 = 0\) → \(x = 12\), or \(x + 9 = 0\) → \(x = -9\).
Step 9: Since a natural number must be positive, we take \(x = 12\).
Final Answer: The number is 12.