Problem 43 - 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 \(f(x)=0.1\) $$\frac{1}{4}\left(x^{2}-10 x+17\right)=0$$

Step-by-Step Solution

Answer

The graph of the equation \(y=x^{2}-10x+17\) intersects the x-axis at two points. The x-coordinates of these points are the solutions to the equation. The exact solutions can be approximated using a graphing utility.

Step 1: Rewrite the Equation

First, the equation needs to be written in the form \(f(x) = 0\). Multiply the equation by 4 so it takes the form \(x^{2}-10x+17 = 0\)

Step 2: Use a Graphing Utility

Next, graph the function \(y=x^{2}-10x+17\). Using a graphing utility, plot this equation on the x-y coordinate axis. The points where the graph cuts the x-axis are the solutions to our equation.

Step 3: Co-ordinates of Cutting points

Determine the x-coordinates of these points of intersection. These x-values are the solutions to the equation.

Key Concepts

Graphing Utility

Graphing utilities are amazing tools that help us visualize equations and functions. They create a visual graph based on the mathematical function you input.
For our equation, transformed into the form \(x^2 - 10x + 17 = 0\), you would plot \(y = x^2 - 10x + 17\) on the graphing utility.

• The graphing utility takes care of calculating the y-values for each x and plots the points accordingly.
• This is especially useful for quadratic equations because it can be tricky to simply visualize their behavior by looking at the equation alone.
• The graph will display a parabola, which either opens upwards or downwards.

This parabola helps you see where the equation intersects the x-axis, which indicates the solutions to the equation. Using a graphing utility can make finding these intersection points quick and straightforward.

Intersection Points

In the context of solving quadratic equations, intersection points on a graph are crucial. They represent the solutions to the equation. When we talk about an intersection point, it’s where the graph of the function crosses the x-axis.
In our re-written equation \(x^2 - 10x + 17 = 0\), this means looking for points where \(y = 0\).

• For a quadratic, these points of intersection are the x-values where the curve meets the axis.
• If you have a graph that doesn’t touch the x-axis at all, this indicates that there are no real solutions, and the quadratic equation has imaginary or complex solutions.
• On a graphing utility, these intersection points can be estimated by identifying where the curve crosses or appears to cross the x-axis.

Understanding intersection points help us decode the graphical representation of solutions, thus solving the quadratic equation.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations algebraically. It applies to any quadratic equation of the form \(ax^2 + bx + c = 0\).
For our equation \(x^2 - 10x + 17 = 0\), the coefficients would be \(a = 1\), \(b = -10\), and \(c = 17\).

• The quadratic formula is given as: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
• This formula will provide the exact solutions (x-values) where the quadratic function equals zero.
• It's particularly useful when plotting the graph because it provides precise intersection points with the x-axis.

Using both this formula and a graphing utility can help confirm solutions and give a clearer picture of the algebraic and graphical solves of a quadratic equation.