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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 unknown positive number is 3.

Step by step solution

01

Translate the problem into equation

Let the positive number be denoted by x. Then according to the problem, the equation can be formulated as: \(x^2 - (6 + 2x) = 0\).
02

Simplify the Equation

Simplify the equation by getting rid of the brackets and combining like terms, we have: \(x^2 - 6 - 2x = 0 \). After rearranging, we have \(x^2 - 2x - 6 = 0\).
03

Solve the Quadratic Equation

Now, we can solve this quadratic equation using the formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. We know that in the equation \(ax^2 + bx + c = 0\), a = 1, b = -2, and c = -6. After substituting these values, we have two solutions: \(x = \frac{2 \pm \sqrt{(-2)^2 - 4*1*(-6)}}{2*1}\). This simplifies to \(x = 3, -2\).
04

Determine the Valid Solution

Since we are looking for a positive number, we reject -2. Therefore, the positive number is 3.

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