Problem 125 - College Algebra

Question

When the sum of 6 and twice a positive number is subtracted from the square of the number, 0 results. Find the number.

Step-by-Step Solution

Answer

The positive number that satisfies the given equation is 3.

Step 1: Set Up the Equation

The problem can be translated into the quadratic equation: \(n^2 - 2n - 6 = 0\). This is achieved by rearranging the terms in the given problem statement.

Step 2: Apply the Quadratic Formula

With this quadratic equation, we identify the coefficients for the formula \(ax^2 + bx + c = 0\) as \(a = 1\), \(b = -2\), and \(c = -6\). Plugging these into the quadratic formula gives two possible solutions: \(n_1 = \frac{-(-2) + \sqrt{(-2)^2 - 4(1)(-6)}}{2(1)}\) and \(n_2 = \frac{-(-2) - \sqrt{(-2)^2 - 4(1)(-6)}}{2(1)}\). Simplifying this results in \(n_1 = 3\) and \(n_2 = -2\).

Step 3: Select the Correct Solution

The problem statement specifies that we are looking for a positive number. Therefore, \(n_2 = -2\) is not a valid answer to this problem because it's not positive. Thus, we are left with only one solution, which is \(n_1 = 3\).