Problem 2 Solve each equation. $$ (x+3... [FREE SOLUTION]

Chapter 6: Problem 2

Solve each equation. $$ (x+3)(x+2)=0 $$

Short Answer

Expert verified

The solutions are $x=-3$ and $x=-2$.

Step by step solution

Set Each Factor to Zero

The given equation is $(x+3)(x+2)=0$. Since the equation is in factored form, we use the zero product property, which states that if the product of two expressions is zero, then at least one of the expressions must be zero. Therefore, we set each factor equal to zero: $x+3=0$ and $x+2=0$.

Solve the First Equation

For the first equation $x+3=0$, subtract 3 from both sides to solve for $x$. This gives: $x=-3$.

Solve the Second Equation

For the second equation $x+2=0$, subtract 2 from both sides to solve for $x$. This gives: $x=-2$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Zero Product Property

The zero product property is a fundamental principle in algebra that makes solving equations much more straightforward. It states that if the product of two or more factors is zero, then at least one of the factors must be zero. This is particularly useful when working with quadratic or higher degree polynomials.

Imagine you are multiplying several numbers and the result is zero. This only happens when at least one of the numbers is zero.
In the context of the equation \[(x+3)(x+2)=0\] we use this property to break down the equation into simpler parts by setting each factor equal to zero.

This process leads us directly to the potential solutions of the equation. Understanding the zero product property is crucial for solving many algebraic problems efficiently.

Factored Form

Factored form refers to a mathematical expression rewritten as a product of its factors. It's like breaking down a number or an expression into components that, when multiplied together, give back the original expression.

For example, in the expression \[(x+3)(x+2)\] we have it broken down into two factors: $x+3$ and $x+2$.
This is particularly useful because, once in factored form, we can easily apply the zero product property to solve equations.

Factoring can simplify solving equations and is a critical skill in algebra. By recognizing and utilizing the factored form, students can approach problems in a more manageable way.

Linear Equations

Linear equations are equations that form a straight line when graphed. They have variables raised only to the power of one and are one of the simplest types of equations to solve in algebra.

In the context of setting $x+3=0$ and$x+2=0$, we solve each like a linear equation to find the possible values of $x$.
Solving linear equations typically involves simplifying the equation through basic arithmetic operations like addition, subtraction, multiplication, or division.

The solutions found, $x=-3$ and $x=-2$, are straightforward because they require only one operation to isolate $x$. Understanding linear equations is a foundation that helps in solving more complex algebraic expressions efficiently. This concept often serves as an introduction to more advanced topics in algebra.

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91影视

Solve each equation. $$ (x+3)(x+2)=0 $$

Short Answer

Step by step solution

Set Each Factor to Zero

Solve the First Equation

Solve the Second Equation

Key Concepts

Zero Product Property

Factored Form

Linear Equations

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