Problem 126 - College Algebra Essentials

Question

When the sum of 1 and twice a negative number is subtracted from twice the square of the number, 0 results. Find the number.

Step-by-Step Solution

Answer

The number is -0.5

Step 1: Translation to Algebraic Equation

Translate the given problem into a mathematical equation. Use \(x\) as the unknown number. The problem can be rewritten as \(2x^2 - (1 + 2x) = 0\).

Step 2: Solving the quadratic equation

Next, the quadratic equation needs to be solved. Simplify the equation to make it in the standard form of a quadratic equation \(ax^2 + bx + c = 0\). Here, the equation after simplifying becomes \(2x^2 + 2x - 1 = 0\).

Step 3: Finding the roots using Quadratic formula

To find the roots of the equation \(2x^2 + 2x - 1 = 0\), use the quadratic formula, which is \(x = {-b \pm \sqrt{b^2-4ac}} / {2a}\). On substitution for \(a\), \(b\), and \(c\), we get \(x = {-2 \pm \sqrt{(2)^2 - 4*2*(-1)}} / {2*2}\). Therefore, the roots \(x\) are \(-0.5\) and \(1\). But since the problem specifies that the number is negative, the number is \(-0.5\).