Problem 4 - Algebra 1: Concepts and Skills

Question

Find and correct the error at the right. \(\begin{aligned}(2 x+4)(x-2) &=0 \\ 2 x+4 &=0 \quad \text { or } \quad x-2=0 \\\ 2 x &=4 \\ x &=2 \end{aligned}\)

Step-by-Step Solution

Answer

The correct solutions for the equation are \( x = -2 \) and \( x = 2 \). The error in the original exercise occurred during simplification of the first part (factor), resulting in an incorrect solution for x.

Step 1: Equation Overview

The equation given is \( (2 x+4)(x-2) = 0 \). To solve this equation, each of the factors need to be set equal to zero and solved for x.

Step 2: Identify and Correct the Error

The next step is \( 2x + 4 = 0 \) or \( x - 2 = 0 \). When the first part is simplified, it results in \( 2x = 4 \), which gives \( x = 2 \). However, the actual simplification should be: Subtracting 4 from both sides gets \( 2x = -4 \), then dividing both sides by 2 gives \( x = -2 \). The second part \( x - 2 = 0 \) is solved correctly, with \( x = 2 \).

Step 3: Statement of Correct Solutions

The correct solutions are \( x = -2 \) or \( x = 2 \).

Key Concepts

Factoring

Factoring a polynomial involves breaking it down into simpler expressions that multiply together to give the original polynomial.
This is especially useful when solving quadratic equations. In this exercise, we are given the equation \((2x+4)(x-2) = 0\). It is already factored, which makes application of the Zero Product Property straightforward.

When factoring, look for common factors that can be factored out of each term. If the polynomial is not already factored, you would often start by factoring out the greatest common factor (GCF).

In some cases, you might use techniques like splitting the middle term or applying special product formulas, such as the difference of squares. This factorization helps in setting up the equation for the next step of solving using the Zero Product Property.

Zero Product Property

The Zero Product Property is a crucial concept in solving quadratic equations that have been factored.
It states that if the product of two expressions is zero, then at least one of the expressions must be zero: \(A \times B = 0\) means either \(A = 0\) or \(B = 0\).
This property simplifies the process of solving equations, as shown with the equation \((2x+4)(x-2) = 0\).

To apply the Zero Product Property, set each factor equal to zero:

• \(2x + 4 = 0\)
• \(x - 2 = 0\)

Solving these equations will yield the values of \(x\) that make the original equation true. The Zero Product Property is a powerful tool for identifying these solutions quickly and accurately.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true.
For quadratic equations, this often means applying methods like factoring and using the Zero Product Property.

In our exercise, after factoring the equation and using the Zero Product Property, we end up with two simple equations:

• \(2x + 4 = 0\)
• \(x - 2 = 0\)

For the first equation, correct simplification gives:

• Subtract 4 from both sides: \(2x = -4\)
• Divide both sides by 2: \(x = -2\)

For the second equation, solving gives:

• \(x = 2\)

This straightforward approach leads us to the correct solutions for the equation: \(x = -2\) or \(x = 2\).
Practice with multiple problems to build confidence in each step, ensuring accuracy in finding solutions.