Problem 48 Solve each equation. See Example... [FREE SOLUTION]

Chapter 6: Problem 48

Solve each equation. See Examples I through 9. (A few exercises are linear equations.) $$ x^{2}+15 x=0 $$

Short Answer

Expert verified

The solutions to the equation are $ x = 0 $ and $ x = -15 $.

Step by step solution

Identify the Type of Equation

Recognize that the given equation $ x^2 + 15x = 0 $ is a quadratic equation. Quadratic equations are in the form $ ax^2 + bx + c = 0 $. Here, $ a = 1 $, $ b = 15 $, and $ c = 0 $.

Factor the Equation

Since the equation can be factored, write it as a product of terms. Factor out the common variable $ x $:\[ x(x + 15) = 0 \]

Solve Each Factor

Set each factor equal to zero and solve for $ x $. This gives us two equations: $ x = 0 $ and $ x + 15 = 0 $.

Solve for Each Solution

Solve the second equation: $ x + 15 = 0 $ becomes $ x = -15 $.

List the Solutions

Combine the solutions from each factor to list all possible values of $ x $. The solutions are $ x = 0 $ and $ x = -15 $.

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Key Concepts

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

Factoring Quadratic Equations

Factoring is a very handy method for solving quadratic equations, especially when they can be simplified into a product of simpler expressions. A quadratic equation like $ x^2 + 15x = 0 $ can be factored easily because it lacks a constant term. This means we can extract a common factor from both terms, making the solution more approachable.Here's how factoring works in this case:

First, recognize that both terms in the equation, $ x^2 $ and $ 15x $, share a common factor, $ x $.
Factor out the $ x $ from the equation, which transforms $ x^2 + 15x = 0 $ into $ x(x + 15) = 0 $.

This process breaks down the equation into simpler parts, making it easier to solve. Factoring is especially useful because once an equation is factored, each factor can be independently set equal to zero to find the solutions.

Solving Equations through Factoring

Once you've factored the equation, you've laid the groundwork for finding solutions. With the equation $ x(x + 15) = 0 $, you have two simpler equations ready to solve: $ x = 0 $ and $ x + 15 = 0 $.To solve these equations, apply these steps:

Set each factor equal to zero. This stems from the zero-product property, which states that if a product of factors equals zero, at least one of the factors must be zero.
The first equation, $ x = 0 $, is already solved.
The second factor, $ x + 15 = 0 $, can be re-arranged to find $ x = -15 $ by subtracting 15 from both sides.

By solving each part, you determine the possible values of $ x $ that satisfy the original quadratic equation. Thus, the solutions are $ x = 0 $ and $ x = -15 $.

Understanding Quadratic Equations in Algebra

Quadratic equations are a fundamental component of algebra, characterized by the general form $ ax^2 + bx + c = 0 $. The solutions to these equations are often referred to as 'roots' or 'zeros'.Here are some important aspects:

Quadratics can often be solved through several methods including factoring, using the quadratic formula, or completing the square.
Factoring is typically the first method to try when the quadratic can be easily expressed as a product of simpler factors.
The degree of the equation (which is 2 for quadratics) indicates the number of solutions you should expect. Therefore, $ x^2 + 15x = 0 $ will have two solutions, which we found to be $ x = 0 $ and $ x = -15 $.

Understanding how to manipulate and solve these equations is crucial for mastering topics in algebra, as these skills form the foundation for more complex mathematics.

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91影视

Solve each equation. See Examples I through 9. (A few exercises are linear equations.) $$ x^{2}+15 x=0 $$

Short Answer

Step by step solution

Identify the Type of Equation

Factor the Equation

Solve Each Factor

Solve for Each Solution

List the Solutions

Key Concepts

Factoring Quadratic Equations

Solving Equations through Factoring

Understanding Quadratic Equations in Algebra

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