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Chapter 1: Problem 115

Solve \(x^{2}+2 x=0\).

Short Answer

Expert verified
The solutions are \(x = 0\) and \(x = -2\).

Step by step solution

01

- Factor the equation

Rewrite the quadratic equation in a form that allows factoring. You have: \[x^2 + 2x = 0\]. Factor out the common term, which is \(x\): \[x(x + 2) = 0\].
02

- Apply the Zero Product Property

According to the Zero Product Property, if a product of factors is zero, at least one of the factors must be zero. Thus, set each factor equal to zero: \[x = 0\] \[x + 2 = 0\].
03

- Solve for x

Solve each equation separately: 1. For the equation \(x = 0\), the solution is simply \(x = 0\). 2. For the equation \(x + 2 = 0\), solve for \(x\) by isolating the variable: \[x = -2\].
04

- List all solutions

Combine the solutions from both equations. The solutions to the quadratic equation are \(x = 0\) and \(x = -2\).

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Key Concepts

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

Factoring Quadratic Equations
When dealing with a quadratic equation like \(x^2 + 2x = 0\), the first step is to factor it. The goal is to rewrite the equation so that it becomes a product of simpler expressions.

For example, consider the equation \(x^2 + 2x = 0\). Both terms on the left side have a common factor, which is \(x\). By factoring \(x\) out, we get: \[x(x + 2) = 0\].

This factoring step is crucial because it transforms the equation into a form that is easier to solve. Instead of dealing with a polynomial, you now have a product of linear expressions. Always look for the greatest common factor (GCF) to factor out in such cases.
Zero Product Property
The Zero Product Property is a fundamental principle used in solving quadratic equations. It states that if the product of two factors is zero, at least one of the factors must be zero.

Applying this property to our factored equation \[x(x + 2) = 0\], we set each factor equal to zero: \[x = 0\] \[x + 2 = 0\].

By doing this, we break down the problem into simpler equations that are easier to solve. This property is extremely useful because it allows us to isolate each variable and solve for its value. Remember, if you have a product equal to zero, every individual factor can be examined separately.
Solving for Variables
After applying the Zero Product Property, the next step is solving for \(x\). For our equations: \[x = 0\] \[x + 2 = 0\], we solve each one separately.

  • First equation: \(x = 0\). This is already solved, so \(x\) is 0.
  • Second equation: \(x + 2 = 0\). Subtract 2 from both sides to isolate \(x\): \[x = -2\].

Combining these solutions, we have \(x = 0\) and \(x = -2\). These are the solutions to the original quadratic equation. Solving each simpler equation independently helps manage complex problems and ensures that you don't miss any solutions.

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