Problem 49 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ 3 x^{2}=-8 x $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = -\frac{8}{3}\).

Step 1: Set the Equation to Zero

The given equation is \(3x^2 = -8x\). Our first step is to rearrange this equation so that one side is zero. We do this by adding \(8x\) to both sides: \[3x^2 + 8x = 0\]

Step 2: Factor the Quadratic Expression

We now have the equation \(3x^2 + 8x = 0\). The next step is to factor out the common factor \(x\):\[x(3x + 8) = 0\]

Step 3: Apply the Zero Product Property

According to the zero product property, if a product of factors equals zero, at least one of the factors must be zero. So we set each factor equal to zero: 1. \(x = 0\)2. \(3x + 8 = 0\)

Step 4: Solve Each Equation

Solve each of the equations from Step 3:- The equation \(x = 0\) is already solved.- To solve \(3x + 8 = 0\), subtract 8 from both sides to get \(3x = -8\) and then divide both sides by 3 to find: \[x = -\frac{8}{3}\]

Step 5: List the Solutions

We have solved for \(x\) in both cases: the solutions to the equation \(3x^2 = -8x\) are \(x = 0\) and \(x = -\frac{8}{3}\).

Key Concepts

Factoring Quadratics

Factoring quadratics is a fundamental technique in algebra used to simplify equations, making them easier to solve. The goal is to express a quadratic equation in the form of a product of its factors. This approach helps in finding the solutions or roots of the equation.

In our example equation, \[3x^2 + 8x = 0\]The common technique for factoring is to look for common factors in all terms. Here, both terms share an \(x\). By factoring out this common factor, you reduce the equation to a simpler form:

• Identify the greatest common factor (GCF). For this equation, the GCF is \(x\).
• Factor out \(x\) to rewrite the equation as \(x(3x + 8) = 0\).

By factoring, you simplify the process of solving quadratic equations significantly.

This step is crucial as it paves the way for applying the zero product property efficiently.

Zero Product Property

The zero product property is a key concept when solving quadratic equations. It states that if the product of two factors is zero, then at least one of the factors must be zero.

Once we have factored our equation to:\[x(3x + 8) = 0\]We can apply the zero product property:

• Set each factor equal to zero: \(x = 0\) and \(3x + 8 = 0\).
• If \(x = 0\), this immediately provides a solution: \(x = 0\).
• If \(3x + 8 = 0\), we'll solve for \(x\) to find: \(x = -\frac{8}{3}\).

This property simplifies the process as it turns a quadratic equation into two linear equations, which are much easier to solve.

Understanding and applying the zero product property is vital in algebra, particularly when working with quadratic equations and polynomial expressions.

Step by Step Algebra Solution

Solving problems step by step can break down complex problems into manageable parts, easing overall comprehension. To illustrate, let's look through our exercise:

1. **Set Equation to Zero**
Add terms to one side to ensure the equation equals zero:\[3x^2 + 8x = 0\]2. **Factor the Quadratic**
Identify the common factor, here being \(x\), and factor it out:\[x(3x + 8) = 0\]3. **Apply the Zero Product Property**
Resolve each factor set to zero:

• First factor: \(x = 0\)
• Second factor: Solve \(3x + 8 = 0\) to find \(x = -\frac{8}{3}\)

The step-by-step method guides you through each logical phase, ensuring no steps are missed.

By digesting each part of the solution, you can handle similar problems more confidently and efficiently.