Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate any solutions of the equation. [Remember to write the equation in the form \(f(x)=0.1\) $$-\frac{1}{2}\left(x^{2}-6 x+6\right)=0$$

Step-by-Step Solution

Answer

By plotting the graph, the approximate solutions for the equation can be found at the points where the graph intersects the x-axis. The exact solutions can't be defined without knowing the specific graph.

Step 1: Rewrite the Equation

The equation provided is \(-\frac{1}{2}\left(x^{2}-6 x+6\right)=0\), and if we multiply both sides by -2, we can simplify it to the required form \(f(x) = 0\). Thus, the modified equation becomes \(x^{2}-6x+6 = 0\).

Step 2: Graph the Function

Now draw the graph using the given equation \(x^{2} - 6x + 6\), using a graphing tool. The graph would likely be a parabola opening upwards because of the positive coefficient of \(x^2\). Our interest points are where this graph intersects the x-axis, as these are the real roots of the equation.

Step 3: Identify Intersection Points

Look for the points where the graph intersects the x-axis, as these points are the solutions for the equation. Depending on the nature of the graph, there might be two, one, or no real intersections, indicating two, one, or no real roots respectively.

Key Concepts

Graphing Utilities

In today's digital age, graphing utilities have become indispensable tools for students learning about quadratic equations. These utilities can be software programs or handheld calculators that assist in graphing complex equations with ease. They help visualize problems that are often abstract when represented in only numerical form.

Using a graphing utility is advantageous because:

• They offer a precise graphical representation of equations.
• They allow easy manipulation of equations to see immediate results.
• They can identify key features such as roots, intercepts, and vertex with simple commands.

For the given equation, using a graphing utility helps us plot the equation promptly, which lets us see how it behaves visually. By plotting the function, students can easily discern where the graph intersects the x-axis. This visual aid simplifies identifying real roots, making the abstract numbers more tangible and comprehensible.

Parabolas

The graph of a quadratic equation is known as a parabola. Parabolas are u-shaped curves that can open upwards or downwards. Whether a parabola opens upwards or downwards depends on the coefficient of the squared term in the quadratic equation. If positive, the parabola opens upwards, and if negative, it opens downwards.

For instance, the equation \(x^2 - 6x + 6\) has a positive coefficient for \(x^2\), indicating that the parabola will open upwards. Key characteristics of parabolas include:

• The vertex, which is the highest/lowest point on the parabola.
• The axis of symmetry, a vertical line that runs through the vertex and divides the parabola into two symmetrical halves.
• The intersections with the x-axis, called the roots or solutions of the equation.

This knowledge allows one to understand the structure and form of quadratic equations just by looking at the formula, aiding in solving problems more efficiently.

Real Roots

Identifying real roots is a crucial aspect in understanding the solutions of a quadratic equation. Real roots refer to the x-values where the parabola touches or crosses the x-axis. These values are significant as they represent potential solutions to the equation.

• If the parabola crosses the x-axis twice, there are two distinct real roots.
• If it just touches the x-axis once (at its vertex), there is exactly one real root, known as a repeated or double root.
• If it doesn't intersect the axis, the equation has no real roots.

By employing methods such as graphing or quadratic formula, one can ascertain whether the equation's roots are real and how many there are. For the equation \(x^2 - 6x + 6 = 0\), graphing helped in visually identifying the exact points of intersection with the x-axis, revealing whether these roots are real and how to approximate them accurately.