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Chapter 6: Problem 10

Solve. $$ (2 m+3)(m+8)=0 $$

Short Answer

Expert verified
m = -3/2 and m = -8.

Step by step solution

01

Understand the Zero Product Property

The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, if (2m + 3)(m + 8) = 0, then either 2m + 3 = 0 or m + 8 = 0.
02

Solve for m in the First Equation

Set the first factor equal to zero and solve for m:2m + 3 = 0Subtract 3 from both sides:2m = -3Divide both sides by 2:m = -3/2
03

Solve for m in the Second Equation

Set the second factor equal to zero and solve for m:m + 8 = 0Subtract 8 from both sides:m = -8
04

State the Solutions

The solutions to the equation (2m + 3)(m + 8) = 0 are m = -3/2 and m = -8.

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Key Concepts

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

Solving Quadratic Equations
Solving quadratic equations is a fundamental skill in algebra. A quadratic equation is typically of the form . In some cases, quadratic equations can be set up as products of binomials, such as \((2m + 3)(m + 8) = 0\). Understanding how to solve these can help in numerous mathematical problems. The key method here is to use the Zero Product Property.
Factoring
Factoring is rewriting an equation as a product of simpler expressions. It's a useful tool for simplifying problems or solving equations. For example, \((2m + 3)(m + 8) = 0\) is already factored. By breaking down complex expressions into products of factors, you can use properties like the Zero Product Property to find solutions more easily.
Solutions to Algebraic Equations
To find solutions to algebraic equations like \((2m + 3)(m + 8) = 0\), you need to find the values of the variable that make the equation true. With the Zero Product Property, if a product equals zero, at least one of the factors must be zero.
So, we set each factor to zero:
  • For \(2m + 3 = 0\), solve by: subtracting 3:\(2m = -3\) then dividing by 2:\(m = -\frac{3}{2}\)

  • For \(m + 8 = 0\), solve by: subtracting 8:\(m = -8\)
Basic Algebra Concepts
Basic algebra concepts are foundational for solving equations. These include understanding variables, constants, and coefficients, along with operations like addition, subtraction, multiplication, and division.
In the given equation \((2m + 3)(m + 8) = 0\), we applied these basic operations to isolate the variable \(m\) and find its values. Mastering these skills is crucial for advancing in mathematics.
Using these methods consistently will improve your problem-solving abilities and confidence in handling algebraic expressions.

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Most popular questions from this chapter

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Solve. $$ (x-4)(x-4)=0 $$

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