Problem 125 - College Algebra Essentials

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 solutions to the problem are \(x = 1 + sqrt{7}\) and \(x = 1 - sqrt{7}\), but since x has to be a positive number, the solution must be \(x = 1 + sqrt{7}\), which approximates to 3.65

Step 1: Translation to Equation

The problem can be translated into the following quadratic equation: \(x^2 - (6 + 2x) = 0\). Where x is the positive number that the problem is asking to find.

Step 2: Simplifying Equation

After simplifying the above equation, it becomes: \(x^2 - 2x - 6 = 0\)

Step 3: Solving the Quadratic Equation

By using the quadratic formula \(x = [-(-2)±sqrt{(-2)^2 - (4)(1)(-6)}]/2(1)\). This simplifies further to \(x = (2±sqrt{4 + 24})/2\), which gives us two possible solutions: \(x = (2±sqrt{28})/2 = 1±sqrt{7}\)