Problem 43 - Intermediate Algebra

Question

Solve each quadratic equation using the method that seems most appropriate. $$ (3 x-1)(2 x+9)=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{3}\) and \(x = -\frac{9}{2}\).

Step 1: Apply the Zero Product Property

To solve the quadratic equation \[(3x - 1)(2x + 9) = 0\],you can use the Zero Product Property. The property states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two separate equations to solve:1. \(3x - 1 = 0\) 2. \(2x + 9 = 0\)

Step 2: Solve the First Equation

Start with the equation \(3x - 1 = 0\).To isolate \(x\), add 1 to both sides: \(3x = 1\).Then, divide both sides by 3:\(x = \frac{1}{3}\).

Step 3: Solve the Second Equation

Now solve the equation \(2x + 9 = 0\).Subtract 9 from both sides to get:\(2x = -9\).Finally, divide both sides by 2:\(x = -\frac{9}{2}\).

Step 4: List All Possible Solutions

After solving both separate equations, we find the solutions\(x = \frac{1}{3}\) and \(x = -\frac{9}{2}\). These are the roots of the original quadratic equation.

Key Concepts

Zero Product Property

The Zero Product Property is a fundamental concept in algebra that simplifies solving quadratic equations. When you have a product set equal to zero, like \((3x - 1)(2x + 9) = 0\), this property can be applied. It's based on the logical idea that if two multiplied factors result in zero, at least one of those factors must be zero itself.
This means you can break the equation into two easier-to-solve parts. In our case:

• \(3x - 1 = 0\)
• \(2x + 9 = 0\)

Using this property is efficient because it allows you to solve quadratic equations by addressing simpler, linear equations. Understanding this idea of breaking down problems can make complex equations less intimidating and more manageable.

Factoring

Factoring is another powerful tool often utilized in solving quadratic equations. It involves expressing an equation as a product of its factors. In terms of our earlier example,\((3x - 1)(2x + 9)\), these are the equation's factors.
When a quadratic equation is presented in a factored form, like our example, it becomes straightforward to apply the Zero Product Property. Factoring a more complicated quadratic expression such as \(ax^2 + bx + c\) requires recognizing patterns, such as:

• Common factor: Simplifying expressions by factoring out common elements.
• Difference of squares: Recognizing patterns such as \(a^2 - b^2\).
• Trinomial squares: An expression that looks like \((x + d)^2\).

Recognizing these patterns means you're well-equipped to factor even more complex equations later on. This readiness transforms intimidating tasks into simple arithmetic.

Roots of Equations

The concept of 'roots of equations' is central in understanding the solutions to quadratic equations. The roots are the values of \(x\) that satisfy the equation \((3x - 1)(2x + 9) = 0\). In this context, solving the equation gives us the roots \(x = \frac{1}{3}\) and \(x = -\frac{9}{2}\).
These roots represent the points where the graph of the quadratic equation touches or crosses the x-axis on a coordinate plane.
Finding the roots of a quadratic equation helps in understanding how the equation behaves graphically. Roots can be:

• Real and distinct: Two separate points where the graph intersects the x-axis.
• Real and repeated: A single point where the graph just touches the x-axis.
• Complex: When the parabola doesn’t intersect the x-axis, leading to solutions that involve imaginary numbers.

Exploring these different conditions helps you visualize and interpret quadratic functions better.