Problem 75 - Algebra A Combined Function

Question

Solve each equation. \((x-6)(2 x+1)=0\)

Step-by-Step Solution

Answer

x = 6 or x = -1/2.

Step 1: Apply the Zero Product Property

The equation \((x-6)(2x+1)=0\) uses the zero product property. This property states that if the product of two factors is zero, at least one of the factors must be zero. So, set each factor equal to zero to solve for \(x\). This gives us: 1. \(x-6=0\) 2. \(2x+1=0\).

Step 2: Solve the First Equation

Solve the equation \(x - 6 = 0\) to find one of the possible solutions: Add 6 to both sides of the equation to isolate \(x\): \[ x = 6 \].

Step 3: Solve the Second Equation

Solve the equation \(2x + 1 = 0\) to find the other possible solution: Subtract 1 from both sides: \(2x = -1\). Next, divide both sides by 2: \[ x = -\frac{1}{2} \].

Step 4: Write the Solution Set

Combine both solutions to form the solution set for the equation. The solutions are: \(x = 6\) and \(x = -\frac{1}{2}\). Thus, the solution set is \(\{6, -\frac{1}{2}\}\).

Key Concepts

Zero Product Property

The Zero Product Property is a fundamental concept in algebra that applies when you multiply two numbers, or expressions, together. It states that if the product of these two factors is zero, then at least one of the factors must be zero. This concept is pivotal when dealing with quadratic equations like

• \((x-6)(2x+1)=0\)

In this example, we can set each factor equal to zero:

• \(x-6 = 0\)
• \(2x+1 = 0\)

By doing so, we simplify our task to two separate linear equations. Solving for \(x\) in each will give us the potential solutions for the equation. This process highlights why the zero product property is so useful: it allows us to break down more complex equations into manageable parts.

Solution Set

The solution set is simply the collection of all possible solutions that satisfy the original equation. In our exercise, we used the Zero Product Property to derive two simple equations:

• \(x-6=0\), which simplifies to \(x=6\)
• \(2x+1=0\), which simplifies to \(x=-\frac{1}{2}\)

Thus, the solution set for the given quadratic equation is:

• \(\{6, -\frac{1}{2}\}\)

The solution set is important because it gives you all the values that make the equation true. For any quadratic equation, there can be two, one, or no real solutions, depending on how the equation is set up. In this case, we have exactly two solutions.

Factorization

Factorization is the process of breaking down an expression into a product of simpler components, or 'factors', that when multiplied together give you the original expression. For solving quadratic equations, factorization is a crucial step if applicable.
When you look at the provided equation, \((x-6)(2x+1)=0\), it is already in a factored form. Factorization makes it easier to apply the Zero Product Property, as done in our problem.

Why is factorization important? It simplifies the process of solving quadratic equations by making it possible to apply properties like the Zero Product Property. When an equation is in a factored form, you can quickly find the roots by setting each factor equal to zero, much as we did in this exercise. Factorization turns complex equations into a form that is straightforward and solvable using algebraic methods.