Problem 2 - Intermediate Algebra

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. $$ x^{2}+5 x=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = -5\).

Step 1: Understand the equation

The given equation is \(x^2 + 5x = 0\). We need to find the value of \(x\) that satisfies this equation by factoring.

Step 2: Factor the equation

The expression \(x^2 + 5x\) can be factored by factoring out the common term. Both terms, \(x^2\) and \(5x\), have a common factor of \(x\). So, we factor \(x\) out: \(x(x + 5) = 0\).

Step 3: Apply the Zero Product Property

According to the zero product property, if the product \(a\cdot b\) is zero, then either \(a=0\) or \(b=0\). Here, this means \(x = 0\) or \(x + 5 = 0\).

Step 4: Solve the resulting equations

We now solve the equations from the previous step individually. 1. \(x = 0\) is already solved.2. Solve \(x + 5 = 0\): Subtract 5 from both sides to get \(x = -5\).

Step 5: Final Solution

The solutions to the quadratic equation \(x^2 + 5x = 0\) are \(x = 0\) and \(x = -5\).

Key Concepts

Factoring

Factoring is a technique used to simplify and solve quadratic equations. Imagine breaking down a complex expression into parts that are easier to work with. In mathematics, factoring involves finding common factors or components of an expression and expressing the original equation as a product of these factors.

To factor a quadratic equation like the one given, x^2 + 5x = 0\, we first look for common elements in the terms. In this equation, both terms, \(x^2\) and \(5x\), share a common factor, which is \(x\).

Once identified, we "factor out" the common \(x\), resulting in the equation \(x(x + 5) = 0\). This process makes the equation much simpler to solve, as it breaks it down into more manageable parts.

Zero Product Property

The zero product property is a fundamental principle for solving equations, particularly when dealing with factored forms.

This property states that if a product of two numbers \((a)\times(b)\) equals zero, then at least one of the numbers \(a\) or \(b\) must be zero. In simpler terms, if \(ab = 0\), then either \(a = 0\) or \(b = 0\). This is because zero is a unique number that, when multiplied by any other number, results in zero.

In our example, the factored equation is \(x(x + 5) = 0\). By applying the zero product property, we find that either \(x = 0\) or \(x + 5 = 0\). This allows us to split the equation into separate, simpler linear equations that are easy to solve:

• \(x = 0\)
• \(x + 5 = 0\)

Solving these will give us the possible solutions for the original quadratic equation.

Quadratic Equations

Quadratic equations are polynomial equations of degree two. They have the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.

In our example, the quadratic equation \(x^2 + 5x = 0\) is already simplified with \(c = 0\), making it easier to solve.

Solving quadratic equations can often involve factoring, completing the square, or using the quadratic formula. However, in this case, factoring was ideal because the equation had a common factor that could be easily factored out. Breaking the quadratic equation into simpler components through factoring often leads directly to the solutions we seek, particularly when combined with principles like the zero product property. These techniques transform complex-looking problems into ones we can solve with straightforward logic and algebraic manipulation. The solutions, \(x = 0\) and \(x = -5\), lie at the heart of this approachable yet powerful mathematical tool.