Problem 58 - Applied Calculus

Question

Solve each equation using a graphing calculator. [Hint: Begin with the window [-10,10] by [-10,10] or another of your choice (see Useful Hint in the Graphing Calculator Basics appendix, page A2) and use ZERO or TRACE and ZOOM IN.] $$ 3 x^{2}-6 x+5=2 $$

Step-by-Step Solution

Answer

The solution to the equation 3x^2 - 6x + 3 = 0 is x = 1.

Step 1: Write the equation in standard form

We'll start by writing the given equation in standard form. The original equation is \(3x^2 - 6x + 5 = 2\). To bring it to standard form, move all terms to one side to set the equation equal to zero: \(3x^2 - 6x + 5 - 2 = 0\). Simplify to get: \(3x^2 - 6x + 3 = 0\).

Step 2: Set up the function for graphing

Now that we have the equation \(3x^2 - 6x + 3 = 0\), set up the function \(f(x) = 3x^2 - 6x + 3\). We will graph this function to find its intersections with the x-axis, which represent the solutions to the equation.

Step 3: Choose an appropriate viewing window

Using your graphing calculator, start with a viewing window of [-10,10] by [-10,10]. This means the x-axis will range from -10 to 10, and the y-axis will also range from -10 to 10. Adjust the window if necessary to ensure the entire graph of \(f(x)\) is visible.

Step 4: Graph the equation

Enter the function \(f(x) = 3x^2 - 6x + 3\) into your graphing calculator and plot the graph. Observe where the graph intersects the x-axis, as these points correspond to the solutions of the equation.

Step 5: Find x-intercepts

Use the ZERO or TRACE function on your calculator to accurately locate the x-intercepts of the graph. These x-intercepts are the solutions to the equation \(3x^2 - 6x + 3 = 0\).

Step 6: Record the solutions

After using the ZERO or TRACE function, identify the x-values of the points where the graph intersects the x-axis. These are your solutions for the equation. Suppose the x-intercepts are, for example, \(x = 1\). Verify the intercepts to ensure they are correct.

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation that take the general form \(ax^2 + bx + c = 0\). In these equations, \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). The equation contains a quadratic term \(ax^2\), a linear term \(bx\), and a constant term \(c\).
Quadratics can have up to two solutions, which may be real or complex numbers. To understand this better:

• **Real solutions** occur when the graph of the quadratic equation intersects the x-axis.
• **Complex solutions** arise when there is no intersection with the x-axis, indicating that the solutions have imaginary components.

In this exercise, our quadratic equation \(3x^2 - 6x + 5 = 2\) is rearranged into standard form as \(3x^2 - 6x + 3 = 0\). This form facilitates solving by isolating the problem to finding the x-values where the graph intersects the axis, commonly approached through methods like factoring, completing the square, or using the quadratic formula.

X-Intercepts

An x-intercept is the point where a graph crosses the x-axis. In terms of quadratic equations like our example \(3x^2 - 6x + 3 = 0\), x-intercepts correspond to the solutions of the equation. These are the points where the value of \(y\) becomes zero.
To find x-intercepts:

• **Graph the equation:** Set the function to \(f(x) = 3x^2 - 6x + 3\) and plot it on a graph.
• **Identify the intercept points:** Look where the curve intersects the x-axis. These intersection points represent the solutions \(x\) of the equation.
• **Use graphing calculator functionality:** Utilize features like ZERO or TRACE to get precise x-intercept values.

Remember, there's no guarantee of real solutions; the vertex could lie above the x-axis without intersection if the discriminant \(b^2 - 4ac\) is negative. This would imply no real x-intercepts are present.

Graphing Techniques

Graphing is a powerful visual technique to solve quadratic equations and identify x-intercepts. Here's how to effectively use graphing methods:

• Start by defining the function: Transform your equation into a function like \(f(x) = 3x^2 - 6x + 3\).
• Choose a suitable graphing window: Initially, a window size of [-10,10] by [-10,10] is recommended, but it may need adjustment for visibility.
• Plot the function: Input the function into a graphing calculator and observe its curve.
• Locate x-intercepts: Employ functions such as ZERO or TRACE on your calculator to pinpoint where the curve crosses the x-axis accurately.
• Zoom and adjust: Use ZOOM IN/OUT features to refine views, especially if x-intercepts are near the edges of your screen.

These techniques help provide clear insights into solutions and the behavior of quadratic functions. The graphing calculator is an essential tool that enhances understanding and accuracy when finding real x-intercepts of quadratics.