Problem 3 - Algebra for College Students

Question

Solve each of the quadratic equations by factoring and applying the property, \(a b=0\) if and only if \(a=0\) or \(b=0\). If necessary, return to Chapter 3 and review the factoring techniques presented there. $$x^{2}=-3 x$$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = -3\).

Step 1: Rewrite the Equation

Start with the given equation \(x^2 = -3x\). To use the factoring method, bring all the terms to one side of the equation to set the equation to zero. This transforms the equation to \(x^2 + 3x = 0\).

Step 2: Factor the Equation

Factor the equation \(x^2 + 3x = 0\). Look for the greatest common factor (GCF) in both terms. The GCF here is \(x\). Therefore, write the equation as \(x(x + 3) = 0\).

Step 3: Apply the Zero Product Property

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. Set each factor equal to zero: \(x = 0\) and \(x + 3 = 0\).

Step 4: Solve Each Equation

Solve the simple equations obtained from the previous step. The first equation \(x = 0\) is already solved. For the second equation \(x + 3 = 0\), solve for \(x\) by subtracting 3 from both sides to get \(x = -3\).

Key Concepts

Factoring Techniques

Factoring is a crucial skill in solving quadratic equations. It involves breaking down an expression into a product of simpler expressions. This is akin to disassembling something complex to see how it works. In our example, after rewriting the equation as \(x^2 + 3x = 0\), we notice that both terms have a common factor, which is \(x\). By factoring out this greatest common factor (GCF), we simplify the equation to \(x(x + 3) = 0\).

Different techniques exist for factoring, but finding the GCF is often the most straightforward. When dealing with quadratic expressions, always begin by looking for a GCF. If none exists, you may need to use other methods, such as factoring by grouping or the ac method. Recognizing the appropriate technique comes with practice and familiarity with different types of quadratic expressions.

Zero Product Property

The zero product property is a fundamental concept that facilitates solving quadratic equations after factoring. It states that if a product of factors equals zero, then at least one of the factors must also be zero. This property allows us to solve quadratic equations efficiently once they are factored.

In the equation \(x(x + 3) = 0\), we apply the zero product property by setting each factor to zero: \(x = 0\) and \(x + 3 = 0\). This gives us two simple equations to solve, which leads directly to the solutions of the original quadratic equation. By understanding and using this property, solving equations becomes a step-by-step process where we break down complex equations into solvable parts.

Quadratic Equations

Quadratic equations are mathematical expressions of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. They represent parabolic graphs and have at most two solutions. These solutions can be real or complex numbers, depending on the nature of the equation.

Solving a quadratic equation typically involves manipulating the equation to find its zeros or roots. Techniques include factoring, using the quadratic formula, and completing the square. Each of these methods attempts to make the equation more accessible, ultimately leading to finding where the function value is zero. The example \(x^2 = -3x\) illustrates this by first rewriting as \(x^2 + 3x = 0\), then factoring and solving to find its roots \(x = 0\) and \(x = -3\). Understanding what quadratic equations represent and how to solve them can open up a deeper understanding of algebra and its applications.