Problem 72 - Elementary Algebra

Question

Solve. $$ -3 x(x+8)=0 $$

Step-by-Step Solution

Answer

The solutions are x = 0 and x = -8.

Step 1: Understand the Equation

The equation is \[-3x(x + 8) = 0\].This is a product of two factors that equal zero. According to the zero product property, if the product of two factors is zero, at least one of the factors must be zero.

Step 2: Set Each Factor to Zero

The equation \[-3x(x + 8) = 0\] can be simplified by setting each factor separately to zero: 1. \(-3x = 0\) 2. \(x + 8 = 0\)

Step 3: Solve the First Factor

Solve the first factor \(-3x = 0\).- Divide both sides by -3 to isolate the variable: \[ x = \frac{0}{-3} \]- Therefore, \(x = 0\).

Step 4: Solve the Second Factor

Solve the second factor \(x + 8 = 0\).- Subtract 8 from both sides to isolate the variable: \[ x = -8 \]- Therefore, \(x = -8\).

Step 5: Combine the Solutions

From Steps 3 and 4, we have two solutions: \(x = 0\) and \(x = -8\).These are the values that satisfy the original equation. The solution set is \{0, -8\}.

Key Concepts

Factoring Equations

Factoring equations is a strategy used to simplify and solve equations, especially quadratic equations. It's like breaking down a complex object into simpler parts. In our exercise, the equation \[-3x(x + 8) = 0\] is already factored.

• -3x is one factor
• (x + 8) is the other factor

The key to factoring is to express an equation as a product of simpler terms. This makes solving it easier.
When each factor is set to zero, it’s easier to solve for the unknown variable. Factoring is useful for handling complex expressions and helps in understanding the structure of equations. This method is often used in solving quadratic equations.

Solving Quadratic Equations

Solving quadratic equations often involves bringing the equation into a standard form and using techniques like factoring. In our case, the exercise already presents a factored form of a quadratic equation:\[-3x(x + 8) = 0\]
Once factored, we use the zero product property. This property simplifies solving because if a product of terms is zero, at least one of those terms must be zero.

• If \(-3x = 0\), solve for \(x\) to find one solution.
• If \(x + 8 = 0\), solve for \(x\) to find another solution.

After solving these, we find the values that satisfy the original equation. Quadratic equations often have two solutions, corresponding to each zero of the factors.

Roots of Equations

The roots of an equation are the values of the variable that make the equation true. In the context of our problem:\[-3x(x + 8) = 0\]

• When we solve \(-3x = 0\), we find \(x = 0\).
• When we solve \(x + 8 = 0\), we find \(x = -8\).

These values \(x = 0\) and \(x = -8\) are the roots of the equation. They represent the points where the equation equals zero. Understanding roots is crucial because they represent key points in graphing and analyzing equations.
Roots show us where a curve intersects the x-axis and can illustrate important characteristics of a graph's shape.