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Chapter 1: Problem 125

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

Short Answer

Expert verified
The positive number that satisfies the given equation is 3.

Step by step solution

01

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.
02

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 \(n_2 = -2\).
03

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 \(n_1 = 3\).

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