Problem 47 - College Algebra with Corequisite Support

Question

To solve the quadratic equation \(x^{2}+5 x-7=4,\) we can graph these two equations $$\begin{array}{l} Y_{1}=x^{2}+5 x-7 \\ Y_{2}=4 \end{array}$$ and find the points of intersection. Recall \(2^{\text {nd }} \mathrm{CALC}\) 5:intersection. Do this and find the solutions to the nearest tenth.

Step-by-Step Solution

Answer

The solutions, rounded to the nearest tenth, are approximately \( x = -6.0 \) and \( x = 1.0 \).

Step 1: Write the equations for graphing

We are given two equations to graph: \( Y_1 = x^2 + 5x - 7 \) and \( Y_2 = 4 \). The solutions to the quadratic equation \( x^2 + 5x - 7 = 4 \) are the \( x \)-coordinates of the points where the graphs of these two equations intersect.

Step 2: Set up the equations for graphing

Input these equations into a graphing calculator or graphing software. To do this manually, you would plot \( Y_1 = x^2 + 5x - 7 \) as a parabola and \( Y_2 = 4 \) as a horizontal line.

Step 3: Find points of intersection

With the graph plotted, look for the intersection points of the parabola \( Y_1 \) and the line \( Y_2 \). Use the calculator's intersection feature (often labeled as 2nd CALC > 5:intersection) to accurately determine the \( x \)-coordinates of these points of intersection.

Step 4: Solve and interpret the points of intersection

The calculator will provide one or more \( x \)-values where the graphs intersect. These \( x \)-values are the solutions to the quadratic equation \( x^2 + 5x - 11 = 0 \). Round these solutions to the nearest tenth as instructed.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that can help you visualize mathematical equations. When solving a quadratic equation like \( x^2 + 5x - 7 = 4 \), a graphing calculator can plot the equation and show its graph in an easy-to-understand visual format. This is particularly useful because:

• It allows you to see the graphical representation of the equation.
• It helps you find intersection points quickly.
• It eliminates the need for manual calculation, reducing errors.

To use a graphing calculator, input the equations of the functions you want to analyze. In this case, you would enter \( Y_1 = x^2 + 5x - 7 \) and \( Y_2 = 4 \) to generate their graphs. Once set up, the calculator's display can show both a parabolic curve and a horizontal line. You can then use its built-in features to identify where these graphs cross each other. This visual method drastically simplifies solving equations that could be prone to mistakes if done solely by algebraic manipulation.

Intersection Points

Intersection points of two graphs represent the solutions where their equations hold true at the same time. For the quadratic equation \( x^2 + 5x - 7 = 4 \), you are looking for the \( x \)-coordinates where the parabola \( Y_1 \) and the line \( Y_2 \) meet. This is how to identify those points:

• The graphs crossing tells you that both equations yield the same \( y \)-value at that \( x \)-coordinate.
• The next step is to determine the exact coordinates using the calculator's intersection function.

A graphic feature commonly used is "2nd CALC" followed by "5:intersection," which guides you through selecting curves and navigating to the right intersection. Once done, the calculator provides precise \( x \)-values, ensuring a clear understanding of the solutions.

Parabola and Line Intersection

In this scenario, one of the graphs is a parabola, and the other is a straight line. Here's what that means:

• A parabola generally has a U-shape which can open upwards or downwards depending on the equation, like \( Y_1 = x^2 + 5x - 7 \) in our example.
• The line, \( Y_2 = 4 \), is horizontal, indicating a constant value.

The intersection points are significant because they demonstrate where the value of the quadratic equation \( x^2 + 5x - 7 \) equals 4. By observing the graph, you can determine how many intersection points exist, which translates to the number of solutions for the equation. Visually, it's easier to comprehend scenarios with unique or multiple intersections rather than interpreting algebraic output alone.

Algebraic Solutions

Besides visual methods, algebraic solutions provide an alternative way to solve quadratic equations. To solve \( x^2 + 5x - 7 = 4 \) using algebra:

• First, simplify by setting the equation to zero: \( x^2 + 5x - 11 = 0 \).
• You can apply methods such as factoring, completing the square, or using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

In cases where the quadratic equation cannot be factored easily, the quadratic formula is often the go-to method since it provides exact solutions even when the roots are irrational. Solving algebraically does not require technological tools, but it does demand a solid understanding of algebraic manipulation. Therefore, it's a reliable approach, especially in contexts where calculators are not allowed or practical. This dual approach—graphical and algebraic—empowers students to pick the method that best fits their learning style.