Problem 60 - Introductory Algebra for College Students

Question

Solve each equation and check your solutions. $$x^{3}-4 x=0$$

Step-by-Step Solution

Answer

The solutions for the equation \(x^{3}-4 x=0\) are \(x=0, x=2, x=-2\).

Step 1: Factoring the equation

Factoring the equation \(x^{3}-4 x=0\) gives \(x (x^2 - 4)=0\).

Step 2: Setting each factor to zero

Setting each factor to zero results in two equations: \(x=0\) and \(x^2 - 4=0\). The equation \(x^2 - 4=0\) can be further factored into \((x-2)(x+2)=0\). So the three original equations remain: \(x=0, x-2=0, x+2=0\).

Step 3: Solving for x

Solving each of these equations for x gives: \(x=0, x=2, x=-2\).

Key Concepts

Factoring Equations

Factoring is a powerful technique used to simplify algebraic equations. In the given problem, the cubic equation is initially expressed as \(x^3 - 4x = 0\). Our goal is to break it down into factors that are easier to solve. In essence, this process is about rewriting the equation as a product of simpler expressions. To start factoring this equation, we observe that each term shares a common factor of \(x\). By extracting \(x\), we can rewrite the expression as \(x(x^2 - 4) = 0\). This is the first step in factoring, and it simplifies our equation into a multiplication of simpler parts. Each factor of the equation represents a possible solution when set to zero. Thus, finding these factors lays the groundwork for solving the equation. This technique is essential for tackling algebraic equations, particularly when they involve higher powers like cubes.

Cubic Equations

Cubic equations, such as \(x^3 - 4x = 0\), involve the presence of \(x\) raised to the third power. Understanding these equations begins by recognizing that the highest degree (in this case, three) dictates the curve's general shape and the potential solutions.Solving cubic equations often starts with factoring, as seen in our problem. After factoring \(x\) out, we arrive at \(x(x^2 - 4) = 0\). Solving cubic equations typically results in finding up to three real roots, depending on the equation's structure.In many cubic equations, especially those that appear in homework problems, there may be shortcuts or patterns to observe, such as differences of cubes or sum of cubes, which can make factoring simpler. The method used here effectively demonstrates how factoring reduces the complexity of a problem, breaking it into smaller, manageable pieces, enabling us to find the roots.

Solving for X

After factoring an equation, the next crucial step is solving for the variable \(x\). In our problem, this entails setting each factor of the equation equal to zero. Since any number multiplied by zero is zero, we're looking for the values of \(x\) that satisfy all parts of the factored equation.Firstly, consider the factor \(x = 0\). This is straightforward; setting \(x\) itself to zero satisfies the equation.For the factor \(x^2 - 4 = 0\), we solve for \(x\) by further factoring this quadratic expression as \((x - 2)(x + 2) = 0\). Setting these equal to zero gives us two more solutions: \(x - 2 = 0\) yields \(x = 2\), and \(x + 2 = 0\) gives \(x = -2\).Therefore, the complete solution set for \(x^3 - 4x = 0\) is \(x = 0\), \(x = 2\), and \(x = -2\). Solving for \(x\) involves understanding how factors relate to the solutions and practicing the method of completing and solving simpler equations from one complex expression.