If the square root of a number is subtracted from the number, the result is 272. What is the number?

"If the square root of a number is subtracted from the number, the result is 272. What is the number?"

Step 1: Let the number be $x$

Then the square root of the number is $ \sqrt{x} $.

According to the problem:

$ x - \sqrt{x} = 272 $

Step 2: Substitute $ \sqrt{x} = y $

Let $y = \sqrt{x} \implies x = y^2$.

Substitute in the equation:

$ y^2 - y = 272 $

$ y^2 - y - 272 = 0 $

Step 3: Solve the quadratic equation

Quadratic formula: $ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $

Here, $a = 1, b = -1, c = -272$

$ y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-272)}}{2 \cdot 1} $

$ y = \frac{1 \pm \sqrt{1 + 1088}}{2} $

$ y = \frac{1 \pm \sqrt{1089}}{2} $

$ \sqrt{1089} = 33 $

So, $ y = \frac{1 + 33}{2} = 17 $ (negative solution is ignored)

Step 4: Find the number

$ x = y^2 = 17^2 = 289 $

Answer: 289