Problem 42 - A Graphical Approach to College Algebra

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$x(x-3)=2$$

Step-by-Step Solution

Answer

The solutions are \(x = 2\) and \(x = 1\).

Step 1: Expand the Equation

Begin by expanding the left side of the equation: \(x(x-3)\). This yields \(x^2 - 3x\). So the equation becomes \(x^2 - 3x = 2\).

Step 2: Rearrange the Equation

Next, subtract 2 from both sides to set the equation to zero: \(x^2 - 3x - 2 = 0\).

Step 3: Factor the Quadratic Equation

Now, factor the quadratic equation \(x^2 - 3x - 2 = 0\). We need factors of -2 that add to -3. The factors are \((x - 2)(x - 1)\).

Step 4: Solve for the Roots

Set each factor equal to zero and solve for \(x\): - \((x - 2) = 0\) gives \(x = 2\)- \((x - 1) = 0\) gives \(x = 1\).

Step 5: Graphical Representation

To support the answers graphically, plot the quadratic \(y = x^2 - 3x - 2\) on a graph. The x-intercepts (where \(y = 0\)) will be at \(x = 2\) and \(x = 1\), confirming the real solutions.

Key Concepts

Factoring

Factoring is a critical technique for solving quadratic equations, especially when they are presented in the form of a quadratic trinomial. A quadratic equation is generally expressed as \( ax^2 + bx + c = 0 \). The goal of factoring is to rewrite this equation as a product of two binomials.When solving by factoring, you need to find two numbers that:

• multiply to give the constant term \( c \)
• add to give the middle coefficient \( b \)

In our example, the equation \( x^2 - 3x - 2 = 0 \) was factored into \((x - 2)(x - 1)\). This step involves recognizing the relationship between the product \(-2\) and the sum \(-3\). Getting comfortable with these relationships is key to efficiently factorizing equations.Factoring is not just a tool to simplify equations; it's a powerful method to find solutions that satisfy the equation, known as the "roots." Using factoring helps break down more complex expressions into simpler ones, revealing the underlying structures of the equation.

Graphical Solution

A graphical solution offers a visual method to find the roots of a quadratic equation. It involves plotting the quadratic function on a coordinate plane and observing where the graph intersects the x-axis.For the equation \( y = x^2 - 3x - 2 \), after factoring, we know that the x-intercepts are the solutions to the equation. These intercepts occur where the graph crosses the x-axis, which are the values of \( x \) that make \( y = 0 \). In our case, this happens at \( x = 1 \) and \( x = 2 \).Graphing is quite useful:

• Provides a visual check to confirm the solutions found algebraically.
• Helps understand the nature of the roots (whether they are real or complex, and if they are distinct or repeated).

By observing the parabolic curve intersecting the x-axis, students can visually verify the correctness of the roots obtained through algebraic means. This synergy between algebra and geometry solidifies understanding.

Roots of Equations

The roots of an equation are the values of \( x \) that satisfy the equation \( f(x) = 0 \). These are also the x-intercepts when the equation is graphed. Understanding roots is fundamental as they represent the solutions to the equation in question.For quadratic equations like \( x^2 - 3x - 2 = 0 \), the roots are determined by the factors we found: \( x = 1 \) and \( x = 2 \). These numbers satisfy the original equation, meaning they make the quadratic expression zero.Key points about roots:

• They can be real or complex numbers.
• For the equation \( ax^2 + bx + c = 0 \), the number and type of roots depend on the discriminant \( b^2 - 4ac \).
• Real roots intersect the x-axis, while complex roots do not.

Understanding the roots is crucial for solving quadratic equations, analyzing polynomials, and finding real-world applications. They are integral to both theoretical and practical realms of mathematics.