Problem 34 - Introductory Algebra for College Students

Question

Solve quadratic equation by completing the square. \(3 x^{2}-2 x-4=0\)

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \frac{1}{3} + \frac{√13}{3}\) and \(x = \frac{1}{3} - \frac{√13}{3}\).

Step 1: Rearrange the Equation

First, divide the entire equation by the coefficient of \(x^2\), which happens to be 3 in this case. This step is done to make the coefficient of \(x^2\) equal to 1: \[\frac{3x^2}{3} - \frac{2x}{3} - \frac{4}{3} = 0\], which simplifies to: \(x^2 - \frac{2}{3}x - \frac{4}{3} = 0\)

Step 2: Completing the Square

Arrange the equation such that the numbers are on the right side and the \(x\) terms are on the left. Then add the square of half the coefficient of \(x\) to both sides, that is, let's add \(\left(\frac{\frac{2}{3}}{2}\right)^2 = \frac{1}{9}\) to both sides: \(x^2 - \frac{2}{3}x + \frac{1}{9} = \frac{4}{3} + \frac{1}{9}\). This gives, \(\left(x - \frac{1}{3}\right)^2 = \frac{13}{9}\)

Step 3: Solve for \(x\)

Now, by taking the square root on both sides and considering both the positive and negative square roots, the solutions for \(x\) are gotten as follows: \(x - \frac{1}{3} = ± \frac{√13}{3}\), therefore \(x = \frac{1}{3} ± \frac{√13}{3}\)

Key Concepts

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, which means the highest exponent of the variable is 2. In general, it follows the standard form:

• \(ax^2 + bx + c = 0\), where:
• "\(a\)" is the coefficient of \(x^2\)
• "\(b\)" is the coefficient of \(x\)
• "\(c\)" is the constant term

In our example, the equation given is \(3x^2 - 2x - 4 = 0\). The process involves solving the equation by isolating \(x\). Recognizing quadratic equations is essential, as they appear frequently in various mathematical contexts. It’s important to note that a cannot be zero, as that would make the equation linear, not quadratic.

Techniques to Solve Quadratic Equations

Solving quadratic equations can be approached through different methods. One common method is "completing the square," used in our solution. Here are three main methods:

• Factoring: If the quadratic can be factored into two binomials, it's possible to solve it quickly by setting each factor equal to zero.
• Completing the Square: This involves rearranging the equation and adding a specific value to both sides to create a perfect square trinomial. It's useful in situations where factoring is challenging.
• Quadratic Formula: The formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\) solves any quadratic equation by substituting the values of \(a\), \(b\), and \(c\).

For the provided exercise, completing the square was chosen. This method is particularly helpful when the quadratic equation doesn't easily factorize. It helps transform a complex equation into a simple form that you can solve.

Step-by-Step Solution of Completing the Square

To tackle the equation using the completing the square method, follow these steps:

• Step 1: Simplify the Equation
  Divide the entire equation \(3x^2 - 2x - 4 = 0\) by 3. This helps create a leading coefficient of 1 for \(x^2\). We end with \(x^2 - \frac{2}{3}x - \frac{4}{3} = 0\).
• Step 2: Completing the Square
  Rearrange the equation such that the \(x\) terms are on one side. Add \(\left(\frac{\frac{2}{3}}{2}\right)^2 = \frac{1}{9}\) to both sides to complete the square. The equation now reads: \((x - \frac{1}{3})^2 = \frac{13}{9}\).
• Step 3: Solve for \(x\)
  Take the square root of both sides, remembering to consider both the positive and negative roots. \(x - \frac{1}{3} = \pm \frac{\sqrt{13}}{3}\) gives the solutions: \(x = \frac{1}{3} \pm \frac{\sqrt{13}}{3}\).

Working through each step helps clarify the process and reinforces understanding of the method. Completing the square transforms the quadratic into a manageable form, allowing for precise algebraic solutions.