Problem 48 - Algebra 1: Concepts and Skills

Question

Use a graphing calculator to approximate the solutions of the equation. $$ x^{2}+6 x-7=0 $$

Step-by-Step Solution

Answer

The approximate solutions to the equation \(x^{2}+6x-7=0\) are \(x \approx 1.0\) and \(x \approx -7.0\)

Step 1: Input the equation into the graphing calculator

Key the equation \(x^{2}+6x-7=0\) into the graphing calculator's y= input field.

Step 2: Graph the equation

Press 'GRAPH'. A graph of the function will appear.

Step 3: Find where the graph crosses the x-axis

Use the 'Zero' function (on some calculators, this may be called 'Root' or 'Intersect'). Follow the prompts on your calculator to move the cursor closer to the point where the graph crosses the x-axis, then press 'ENTER'.

Step 4: Record the x-coordinate

The calculator will display the x-coordinate of the point where the graph crosses the x-axis. This is a solution to the equation.

Step 5: Repeat for the other root

If there is another point where the graph crosses the x-axis, repeat Steps 3 and 4 for that point.

Key Concepts

Quadratic Equations

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These are also known as second-degree polynomials because the highest power of the variable \( x \) is 2. Quadratic equations are fundamental in algebra and appear in various real-world contexts.
A quadratic equation illustrates a parabola when graphed, and the shape of the parabola can be concave upwards or downwards, determined by the sign of \( a \).

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

Understanding the form and graph of a quadratic equation helps to predict the behavior of its solutions.

Roots of Equations

The roots of a quadratic equation are the values of \( x \) that satisfy the equation, effectively meaning they make the equation equal to zero. For a quadratic equation like \( x^2 + 6x - 7 = 0 \), the roots represent the x-values where the parabola intersects the x-axis.
Finding the roots can be done in several ways:

• By factoring the quadratic expression, if possible.
• Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Completing the square.
• Graphically, which is the focus of the exercise.

Each method has its strengths, but graphing offers a visual representation which can be very intuitive.

Graphical Method

The graphical method involves plotting the equation on a graph to visually identify its roots. By entering the quadratic equation into a graphing calculator, students can generate a visual depiction of the parabola.
Once plotted, the x-intercepts—points where the graph crosses the x-axis—are the solutions or roots of the equation.

• Using a graphing calculator, you input the equation into the y= field, essentially telling the calculator to graph the function.
• By visually analyzing the graph, you can determine the x-coordinates of the intercepts.
• Modern calculators provide functionality like "Zero" or "Intersect" to assist in pinpointing these locations, making the process efficient and accurate.

The graphical method is particularly helpful for those who learn better visually, providing an alternative to algebraic manipulations.