Problem 51 - College Algebra with Modeling and Visualization

Question

Solve the equation by completing the square. $$ 3 x^{2}-6 x=2 $$

Step-by-Step Solution

Answer

The solutions are \(x = 1 + \sqrt{\frac{5}{3}}\) or \(x = 1 - \sqrt{\frac{5}{3}}\).

Step 1: Move the constant to the other side

The given equation is \(3x^2 - 6x = 2\). The first step is to move the constant on the left side of the equation to the right side by subtracting 2 from both sides. This gives us \(3x^2 - 6x - 2 = 0\).

Step 2: Divide by the coefficient of the quadratic term

Divide the entire equation by 3 to make the coefficient of the \(x^2\) term equal to 1. This gives us \(x^2 - 2x = \frac{2}{3}\).

Step 3: Complete the square

Take the coefficient of the \(x\) term, which is -2, divide it by 2, and square it. This results in \( \left(\frac{-2}{2}\right)^2 = 1\). Add 1 to both sides of the equation to maintain the equality: \(x^2 - 2x + 1 = \frac{2}{3} + 1\).

Step 4: Simplify and solve

The left side of the equation is now a perfect square: \((x-1)^2\). The right side simplifies to \(\frac{5}{3}\), so we have \((x-1)^2 = \frac{5}{3}\).

Step 5: Solve for x

Take the square root of both sides to solve for \(x\). Solving it gives \(x-1 = \pm \sqrt{\frac{5}{3}}\). So, \(x = 1 \pm \sqrt{\frac{5}{3}}\).

Key Concepts

Completing the Square

Completing the square is a method used in algebra to solve quadratic equations. This method transforms the equation into a perfect square trinomial, which can then be easily solved by taking the square root of both sides. Let's break it down with an example:

Consider the quadratic equation:

• Multiply or divide the entire equation by a constant to make the coefficient of the quadratic term (\(x^2\)) equal to 1.
• Take the coefficient of the \(x\) term, divide it by 2, and square the result. This completes the square.
• Add this square to both sides of the equation to maintain balance.

This will structure the left side into a perfect square binomial. For example, in the exercise, after appropriate manipulations, it becomes \((x-1)^2 = \frac{5}{3}\). By setting up the equation in this way, it becomes much simpler to solve.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate variables or make equations easier to solve. This is a fundamental skill in algebra and is crucial when working with more complex expressions, such as quadratics.

The key steps in algebraic manipulation include:

• Moving constants and coefficients from one side of the equation to the other.
• Adding, subtracting, multiplying, or dividing both sides of the equation by the same value to simplify terms.
• Using reverse operations to undo terms, such as taking square roots to solve squared variables.

In solving the equation \(3x^2 - 6x = 2\), the process starts by rearranging terms and making the coefficient of \(x^2\) equal to 1, as seen in the division step \(x^2 - 2x = \frac{2}{3}\). This simplifies the calculation and sets up the equation for completing the square.

Quadratic Equations

Quadratic equations are polynomial equations of degree 2, generally written in the form \(ax^2 + bx + c = 0\). They represent parabolic curves on a graph, and they can be solved using various methods such as factoring, using the quadratic formula, or completing the square.

When given a quadratic equation:

• Identifying the coefficients \(a,\) \(b,\) and \(c\) is the first step.
• Different methods may be more applicable depending on the values of \(a,\) \(b,\) and \(c\). Completing the square is particularly useful when simpler methods like factoring are not applicable.
• Solutions to quadratics can be real or complex numbers, depending on the discriminant \(b^2 - 4ac\).

In the example \(3x^2 - 6x = 2\), we transform it to \(x^2 - 2x = \frac{2}{3}\) before applying the completing the square method, showing a practical application of the theory.