Problem 27 - A Graphical Approach to College Algebra

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$3 x^{2}-2 x=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = \frac{2}{3}\).

Step 1: Identify the equation type

The given equation is a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = 3\), \(b = -2\), and \(c = 0\).

Step 2: Factor the quadratic equation

Rewriting the quadratic equation \(3x^2 - 2x = 0\), we can factor out the common factor \(x\), giving us \(x(3x - 2) = 0\).

Step 3: Solve each factor separately

Set each factor equal to zero, giving the equations \(x = 0\) and \(3x - 2 = 0\). Solve for \(x\): 1. \(x = 0\).2. For \(3x - 2 = 0\), add 2 to both sides to obtain \(3x = 2\). Divide by 3 to find \(x = \frac{2}{3}\).

Step 4: Verify solutions graphically

To support graphically, graph \(y = 3x^2 - 2x\). The solutions correspond to the x-values where the graph intersects the x-axis. If plotted, it shows intersections at \(x = 0\) and \(x = \frac{2}{3}\).

Key Concepts

Factoring a Quadratic Equation

When faced with a quadratic equation, one method to solve it is by factoring. This entails rewriting the quadratic in a form where it can be easily broken down into simpler expressions. In the equation \(3x^2 - 2x = 0\), notice the common factor of \(x\).

Factoring begins by extracting this common factor, transforming the equation into \(x(3x - 2) = 0\). This shows us how the quadratic is composed of two factors.

The reason factoring is so useful is that it allows us to use the zero product property, which states that if the product of two terms is zero, then at least one of the terms must be zero. This simplifies the process of finding the solutions or roots of the quadratic equation.

Always check if there's a common factor before attempting other methods like quadratic formula or completing the square. Factoring can often be quicker and visually more intuitive, too.

Solving Equations Using Factoring

Once a quadratic equation is factored, solving it requires setting each factor to zero. For \(x(3x - 2) = 0\), we set each part of the product to zero individually, leading to two separate equations: \(x = 0\) and \(3x - 2 = 0\).

To solve these equations, start with the simplest one first, \(x = 0\), which gives us one of the solutions immediately.

Next, handle the slightly more complex equation \(3x - 2 = 0\):

• Add 2 to both sides to isolate the term with \(x\): \(3x = 2\).
• Divide both sides by 3 to solve for \(x\): \(x = \frac{2}{3}\).

By solving each factor individually, you find the solutions to the original quadratic equation. Remember, each factor gives rise to a potential solution. Thus, at the end of this process, the solutions are \(x = 0\) and \(x = \frac{2}{3}\).

Graphical Representation of Quadratic Equations

Visualizing a quadratic equation with a graph provides a powerful check of your solutions. By graphing the equation \(y = 3x^2 - 2x\), you can visually confirm the solutions. This is where the graph crosses, or intersects, the x-axis. These intersection points correspond to the values of \(x\) that solve the equation.

To graph the equation:

• Set up a coordinate plane.
• Plot the graph over a range of \(x\) values to see its behavior.

You will notice the graph intersects the x-axis at \(x = 0\) and \(x = \frac{2}{3}\), exactly matching the solutions we found earlier through factoring. This physical representation not only verifies the solutions but also enhances understanding of the quadratic function's behavior.

Remember, while the graph is a helpful tool, always ensure you've done the algebra correctly for a complete solution.