Problem 26 - A Graphical Approach to College Algebra

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=3 x+18$$

Step-by-Step Solution

Answer

The solutions are \(x = 6\) and \(x = -3\).

Step 1: Rewrite the Equation

To solve the equation \(x^2 = 3x + 18\), first rewrite it in a standard quadratic form \(ax^2 + bx + c = 0\). Subtract \(3x + 18\) from both sides to get: \(x^2 - 3x - 18 = 0\).

Step 2: Factor the Quadratic Equation

Now, factor the quadratic equation \(x^2 - 3x - 18 = 0\). Look for two numbers that multiply to \(-18\) and add up to \(-3\). The numbers \(-6\) and \(3\) work, so the equation can be factored as: \((x - 6)(x + 3) = 0\).

Step 3: Solve for x

Set each factor equal to zero and solve for \(x\): 1. \(x - 6 = 0\) gives \(x = 6\).2. \(x + 3 = 0\) gives \(x = -3\). Thus, the solutions are \(x = 6\) and \(x = -3\).

Step 4: Graphical Representation

Graph the function \(f(x) = x^2 - 3x - 18\). The graph of this quadratic function is a parabola that opens upwards. The x-intercepts of the graph are the solutions to the equation, which are \(x = 6\) and \(x = -3\). These points where the parabola crosses the x-axis confirm the solutions graphically.

Key Concepts

Factoring

Factoring is a method used to solve quadratic equations by expressing them as a product of linear factors. Given a quadratic equation in the form \(ax^2 + bx + c = 0\), the goal is to identify two numbers that multiply to \(c\) and add to \(b\). This process transforms the quadratic equation into two simpler linear equations. For instance, in the equation \(x^2 - 3x - 18 = 0\), we need numbers whose product is \(-18\) and sum is \(-3\). The numbers \(-6\) and \(3\) fulfill these requirements.

Thus, the equation can be factored as \((x - 6)(x + 3) = 0\). Factoring helps derive individual solutions by setting each factor to zero, resulting in \(x = 6\) and \(x = -3\). Factoring is often the simplest and quickest method when the quadratic equation is easily factorable.

Quadratic Formula

The quadratic formula provides a reliable way to find the roots of a quadratic equation when factoring is challenging. The formula is expressed as:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Given the equation \(ax^2 + bx + c = 0\), the variables \(a\), \(b\), and \(c\) represent the coefficients. The term \(b^2 - 4ac\) is known as the discriminant and it determines the nature of the roots:

• If the discriminant is positive, there are two real solutions.
• If it is zero, there is one real solution.
• If it is negative, there are no real solutions.

Using the quadratic formula guarantees the solutions to any quadratic equation, making it a powerful tool particularly when simple factoring isn't possible.

Graphing Parabolas

The graphical representation of a quadratic equation is a parabola. This is an important concept because it provides a visual interpretation of the roots or solutions of the equation. In the equation \(f(x) = x^2 - 3x - 18\), when we graph, it creates a parabola opening upwards.

Key characteristics of a parabola:

• Vertex: The highest or lowest point.
• Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
• x-intercepts: Points where the parabola crosses the x-axis, indicating the equation's solutions.

By plotting the function on a graph, we can easily see where it crosses the x-axis. These intercepts are the equation's real solutions, confirming our results from algebraic methods such as factoring or using the quadratic formula.

Real Solutions

Real solutions of a quadratic equation refer to the values of \(x\) that satisfy the equation and are actual, not involving imaginary numbers. When a quadratic equation like \(x^2 - 3x - 18 = 0\) is solved, the solutions are the x-values where the graph intersects the x-axis.

Having real solutions means:

• The discriminant \(b^2 - 4ac\) is non-negative.
• The quadratic equation intersects the x-axis at one or two points. If there is one point, this means the vertex touches the x-axis, indicating one solution.
• If there are two points, it indicates two distinct solutions.

In our case, the solutions \(x = 6\) and \(x = -3\) are real numbers, seen where the parabola intersects the x-axis. Real solutions are crucial in real-world applications where actual values are needed to solve problems.