Problem 24 - Introductory Algebra for College Students

Question

Use factoring to solve each quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\) -intercepts. $$3 x^{2}=5 x$$

Step-by-Step Solution

Answer

The solutions to the equation \(3x^2 = 5x\) are \(x = 0\) and \(x = 5/3\)

Step 1: Set the equation to zero

To start the process, rearrange the equation such that it equals zero. Subtract \(5x\) from both sides of the equation to get: \(3x^2 - 5x = 0\)

Step 2: Factor out the common factor

The left-hand side of the equation has a common factor in both terms which is \(x\). Factor \(x\) out to get: \(x(3x - 5) = 0\)

Step 3: Apply the Zero Product Property

According to the Zero Product Property, if the product of two factors is zero, then at least one of the factors must be zero. So, either \(x = 0\) or \(3x - 5 = 0\).

Step 4: Solve for \(x\)

If \(x=0\), then the first solution for the equation is \(x=0\). For \(3x - 5 = 0\), if we add \(5\) to both sides and then divide by \(3\), we can solve for \(x\). So for \(3x - 5 = 0\), \(x = 5/3\). Therefore, our 2 solutions for \(x\) are \(0\) and \(5/3\)

Key Concepts

Factoring

To begin solving a quadratic equation like \(3x^2 = 5x\), we first need to use factoring effectively. Factoring is the technique of breaking down an expression into simpler terms or expressions that, when multiplied, give the original expression. Here, we want both sides of the equation to equal zero before we factor. Thus, rearrange it to: \(3x^2 - 5x = 0\).

Factoring involves finding a common factor in all the terms of the expression. Essential to classic factoring is identifying factors of a term. In our example, notice that both terms on the left \(3x^2\) and \(-5x\) share a common factor of \(x\). Factoring out \(x\), we rewrite the equation as \(x(3x - 5) = 0\). By doing this, we have expressed the quadratic equation as a product of factors, simplifying our equation for the next step.

Zero Product Property

The Zero Product Property is a fundamental concept when working with equations in factored forms. It states that if a product of two numbers is zero, then at least one of the numbers must be zero.

In mathematical terms, if \(a \cdot b = 0\), then \(a = 0\) or \(b = 0\).

In our factorized equation \(x(3x - 5) = 0\), we apply the Zero Product Property as follows:

• Set each factor equal to zero: \(x = 0\) and \(3x - 5 = 0\).

This approach allows us to solve for each variable independently, deriving the possible solutions for \(x\). The property is particularly useful because it simplifies solving equations by breaking them down into linear, more manageable parts.

Solving Equations

Now that we have our factors set to zero using the Zero Product Property, we can solve the individual equations. Solving these simpler equations gives us the final solutions for the quadratic equation.

Starting with \(x = 0\), this is straightforward as \(x\) is already isolated.
For the equation \(3x - 5 = 0\), we follow these steps:

• Add 5 to both sides: \(3x = 5\).
• Divide both sides by 3 to isolate \(x\): \(x = \frac{5}{3}\).

Thus, the quadratic equation \(3x^2 - 5x = 0\) has two solutions: \(x = 0\) and \(x = \frac{5}{3}\). These solutions signify where the quadratic function would cross the x-axis on a graph, known as the x-intercepts of the function. Therefore, solving quadratic equations via factoring, alongside the Zero Product Property, is an efficient way to find these intercepts.