Problem 43 - Algebra for College Students

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: Identify the Equation Type

The equation given is \((3x-1)(2x+9) = 0\). This is a factored quadratic equation, meaning the quadratic is already expressed as a product of two binomials.

Step 2: Apply the Zero Product Property

According to the zero product property, if the product of two expressions is zero, then at least one of the expressions must be zero. Thus, we set each factor in the equation to zero: \(3x - 1 = 0\) and \(2x + 9 = 0\).

Step 3: Solve the First Equation

Solve \(3x - 1 = 0\) for \(x\):Add 1 to both sides:\[3x = 1\]Divide both sides by 3:\[x = \frac{1}{3}\]

Step 4: Solve the Second Equation

Solve \(2x + 9 = 0\) for \(x\):Subtract 9 from both sides:\[2x = -9\]Divide both sides by 2:\[x = -\frac{9}{2}\]

Step 5: Write the Solution

The solutions to the quadratic equation \((3x-1)(2x+9) = 0\) are \(x = \frac{1}{3}\) and \(x = -\frac{9}{2}\).

Key Concepts

Zero Product Property

The zero product property is a fundamental concept in algebra that helps solve quadratic equations. It states that if the product of two numbers is zero, then at least one of the numbers must be zero.
This property is particularly useful when dealing with equations that are expressed as a multiplication of factors, like quadratic equations in their factored form.
For example, if you have an equation

• \((a)(b) = 0\),

it implies that either:

• \(a = 0\) or
• \(b = 0\).

By setting each factor equal to zero, you can find the possible values for the variable. This is precisely what helps in solving many quadratic equations efficiently.

Factored Quadratic Equation

A factored quadratic equation is one that is expressed as a product of two binomials or factors. Instead of having a standard form like \(ax^2 + bx + c = 0\), it is presented in a product form such as

• \((px + q)(rx + s) = 0\).

This transformation is beneficial because it allows the application of the zero product property.
Factoring is the process used to convert the quadratic into this form. Often, it requires recognizing common factors, using special formulas (like difference of squares), or applying techniques such as grouping. Once factored, solving the equation becomes straightforward. Each binomial can be set to zero, simplifying the solving process.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \(x\) that make the equation true. These values are called 'roots' or 'solutions'. Quadratics can be solved by various methods:

• Factoring
• Completing the square
• Using the quadratic formula

When a quadratic is already factored, as in the given problem, the easiest approach is to apply the zero product property.
You simply set each factor equal to zero and solve for \(x\).
In the original problem, you start with

• \((3x-1)(2x+9) = 0\).

Following the zero product property, set

• \(3x-1 = 0\)

and

• \(2x+9 = 0\)

solving both gives the solutions \(x = \frac{1}{3}\) and \(x = -\frac{9}{2}\). This systematic approach ensures that you find all possible solutions to the quadratic equation.