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Chapter 6: Problem 104

Solve using the principle of zero products. $$ (x+10)(x+11)=0 $$

Short Answer

Expert verified
x = -10 or x = -11

Step by step solution

01

Apply the Principle of Zero Products

The principle of zero products states that if the product of two factors is zero, then at least one of the factors must be zero. Thus, set each factor in the equation \(x + 10\)(x + 11) = 0 to zero: \(x + 10 = 0\) and \(x + 11 = 0\).
02

Solve for x from the First Equation

Solve the first equation \(x + 10 = 0\) for \(x\). Subtract 10 from both sides to get \(x = -10\).
03

Solve for x from the Second Equation

Next, solve the second equation \(x + 11 = 0\) for \(x\). Subtract 11 from both sides to get \(x = -11\).
04

Conclude the Solutions

Putting it all together, the values of \(x\) that satisfy the equation are \(x = -10\) and \(x = -11\).

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Key Concepts

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

Factoring
Factoring is a crucial skill in algebra. It's the process of breaking down a complex expression into simpler 'factors' that you can multiply together to get the original expression. This is often done to make solving equations easier. For example, consider the equation ewline ewline (x+10)(x+11)=0. By setting each term equal to zero, you simplify the problem into two smaller, more manageable equations. Factoring helps identify these terms clearly and is essential for working with quadratic equations and other polynomial forms.
Solving Equations
Solving equations is at the heart of algebra. The goal is to find the unknown variable(s) that make the equation true. When given the factored form ewline (x+10)(x+11)=0ewline, you use the principle of zero products to solve for the variable. This principle states that if a product of two numbers is zero, at least one of the numbers must be zero. Therefore, you can set each factor to zero and solve them individually: ewline 1. (x+10)=0ewline, which gives ewline x=-10, and ewline 2. (x+11)=0ewline, which gives ewline x=-11. Each solution corresponds to a point where the original equation holds true.
Algebraic Principles
Understanding fundamental algebraic principles is essential for solving any math problem effectively. The principle of zero products is a key concept in algebra: ewline If A * B = 0, then either A = 0 or B = 0. This principle helps to break down complex equations into simpler parts. Additionally, knowing how to manipulate equations by adding, subtracting, multiplying, or dividing terms is crucial. Each principle works together to provide a toolkit for solving various types of algebraic expressions more efficiently.

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Most popular questions from this chapter

Write an equivalent expression by factoring out the smallest power of \(x\) in each of the following. $$ x^{-8}+x^{-4}+x^{-6} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 6 x^{2}-6 y^{2} $$

Determine whether each of the following is a difference of squares. $$ -1+64 t^{2} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ (p+q)^{2}-25 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ m^{2}-64 $$
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