Problem 21 - Algebra 1: Concepts and Skills

Question

Use the zero-product property to solve the equation. \((z+2)(z+3)=0\)

Step-by-Step Solution

Answer

The solutions to the equation \((z+2)(z+3) = 0\) are \(z = -2\) and \(z = -3\).

Step 1: Identify Factors

Identify the factors of the equation, which are \((z+2)\) and \((z+3)\).

Step 2: Apply Zero-Product Property

Since the equation \((z+2)(z+3) = 0\) is equal to zero, apply the zero-product property, which states at least one of \((z+2)\) or \((z+3)\) must be zero.

Step 3: Solve the resulting equations

Set each factor equal to zero and solve for \(z\). So, \(z + 2 = 0\) gives \(z = -2\) and \(z + 3 = 0\) gives \(z = -3\).

Key Concepts

Understanding Solving Equations

Solving equations is a fundamental concept in algebra where we find the values of variables that make the equation true. One common type of equation involves expressions that multiply to zero, like \[(z+2)(z+3)=0.\]In this case, the zero-product property simplifies the process.
When solving such equations, we first break down the problem into simpler parts. We identify each factor inside the equation separately and consider their values. The core idea here is to apply properties systematically.

• Write down the equation clearly and understand each part.
• Apply appropriate algebraic rules or properties.
• Search for solutions that satisfy the equation.

The Role of Factoring

Factoring is a crucial step in simplifying equations, especially those set equal to zero. When we factor an equation, we rewrite it as a product of simpler expressions. For the exercise \[(z+2)(z+3)=0,\]we already see the equation in its factored form. This makes it easier to apply mathematical properties and solve it.
Factoring helps in understanding how components of an expression interact. By breaking the expression into factors, we explore different paths to a solution:

• Factorize complex equations into simpler binomials or polynomials.
• Check each factor to determine possible solutions.
• Focus on zero-product property to identify key solutions.

Algebra 1 Concepts Made Simple

Algebra 1 is about learning basic mathematical methods and concepts that build a foundation for more advanced studies. Among these, using the zero-product property and factoring are pivotal techniques. For instance, the equation \[(z+2)(z+3)=0\]exemplifies these concepts beautifully.
The zero-product property simplifies solving equations. It asserts that if a product of multiple factors equals zero, at least one factor must be zero. Therefore, solving the original equation involves simply setting each factor to zero:

• For \(z+2=0\), solve \(z=-2\).
• For \(z+3=0\), solve \(z=-3\).

By systematically tackling such problems, students gain confidence in manipulating algebraic expressions and finding solutions effectively.