Problem 9 - Elementary Algebra

Question

For the following problems, solve the equations. $$ x(2 x+3)=0 $$

Step-by-Step Solution

Answer

Answer: The possible values of x that satisfy the equation are \(x = 0\) and \(x = -\frac{3}{2}\).

Step 1: Utilize the Zero-Product Property

To use the Zero-Product Property, we will set each factor in the equation equal to zero: $$ x = 0 \\ 2x + 3 = 0 $$

Step 2: Solve the first equation: \(x = 0\)

Since x is already isolated, we don't need to perform any additional calculations. Therefore, one of the possible solutions is: $$ x = 0 $$

Step 3: Solve the second equation: \(2x + 3 = 0\)

Now, we'll solve the second equation to find another possible value of x. First, subtract 3 from both sides: $$ 2x = -3 $$ Then, divide both sides by 2: $$ x = -\frac{3}{2} $$

Step 4: Combine the solutions

Now that we've found both values of x that satisfy the given equation, we can present our final solution as: $$ x = 0, -\frac{3}{2} $$

Key Concepts

Zero-Product Property

The Zero-Product Property is a core concept in algebra that is like a special tool for solving equations that involve multiplication. It states that if the product of two numbers or expressions is zero, then at least one of the numbers or expressions must be zero. In more formal terms, if you have an equation where two factors multiply to result in zero, such as \( a \times b = 0 \), then either \( a = 0 \) or \( b = 0 \) or both. This property is very handy because it allows us to break down a complicated equation into simpler parts. For example, when faced with the equation \( x(2x + 3) = 0 \), we apply this property by setting each factor equal to zero: \( x = 0 \) and \( 2x + 3 = 0 \). This transforms one equation into two simpler equations that are easier to solve.

Isolating Variables

Isolating variables is a crucial technique in solving equations. It involves manipulating an equation so that the variable of interest is on one side of the equation on its own. This process makes it easier to identify the value of the variable that satisfies the equation. Taking the equation \( 2x + 3 = 0 \), for instance, our goal is to isolate \( x \). To do this:

• First, subtract 3 from both sides to get rid of the constant term on the side with \( x \): \( 2x = -3 \).
• Next, divide both sides by 2 to solve for \( x \): \( x = -\frac{3}{2} \).

These steps show how simple operations like addition, subtraction, multiplication, or division can help achieve a neatly isolated variable, presenting a clear path to finding its value.

Equation Solutions

Finding the solution to an equation involves finding all possible values of the variable that make the equation true. Once we apply the steps like using the Zero-Product Property and isolating variables, we arrive at potential solutions. For the equation \( x(2x + 3) = 0 \), we followed these processes and discovered two solutions: \( x = 0 \) and \( x = -\frac{3}{2} \). This tells us that plugging these values into the original equation will make it true. Hence, both are valid solutions. This demonstrates that polynomial equations can have multiple solutions, and understanding different properties and techniques is essential in figuring out every possible solution.