Problem 9 - Algebra 1: Concepts and Skills

Question

Solve \(x^{2}-3 x=8\) by completing the square. Solve the equation by using the quadratic formula. Which method did you find easier?

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2} - 3x = 8\) are \(x = \frac{3 + \sqrt{41}}{2}\) and \(x = \frac{3 - \sqrt{41}}{2}\). Both methods 'Completing the Square' and 'Quadratic Formula' provided the same solution. Comparatively, which method is easier tends to be subjective and depends on an individual's preference.

Step 1: 1. Solving by Completing the Square

To solve this equation by completing the square, first, the constant term on the left-hand side needs to be moved to the right-hand side: \(x^{2} - 3x = 8\) becomes \(x^{2} - 3x - 8 = 0\). Now, to complete the square, half the coefficient of 'x', square it, and add it to both sides of the equation: \((x - \frac{3}{2})^{2} = \frac{9}{4} + 8\). Solving through, \(x = \frac{3}{2} \pm \sqrt{\frac{41}{4}}\), which simplifies to \(x = \frac{3 \pm \sqrt{41}}{2}\)

Step 2: 2. Solving using the Quadratic Formula

The quadratic formula is used to solve an equation of the form \(ax^{2} + bx + c = 0\). Our equation \(x^{2} - 3x - 8 = 0\) can be plugged into this formula: \[x = \frac{-(-3) \pm \sqrt{(-3)^{2} - 4(1)(-8)}}{2(1)}\]. Simplifying this, again provides: \(x = \frac{3 \pm \sqrt{41}}{2}\)

Step 3: 3. Evaluating which method was easier

Both methods have provided the same solution, so they're both reliable. The decision on which one is easier can be quite subjective. The 'Completing the Square' method requires you to remember fewer steps, but can be more complex as it involves manipulations like halving, squaring, etc. On the other hand, the 'Quadratic Formula' method involves a simple plug and play, provided you remember the formula correctly, which can be beneficial to solve quickly. Therefore, it's up to individual preference.