Problem 28 - Algebra for College Students

Question

Solve each equation for \(x\) by factoring and applying the property, \(a b=0\) if and only if \(a=0\) or \(b=0\). $$x^{2}+7 k x=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = -7k\).

Step 1: Identify a common factor

Analyze the equation \(x^2 + 7kx = 0\) and identify the common factors of both terms. Notice that both terms contain \(x\) as a factor.

Step 2: Factor out the common factor

Since both terms in the expression \(x^2 + 7kx = 0\) have \(x\) as a common factor, factor \(x\) out of the equation:\[ x(x + 7k) = 0 \]

Step 3: Apply the Zero Product Property

The zero product property states that if \(ab = 0\), then \(a = 0\) or \(b = 0\). Apply this property to the factored equation:1. \(x = 0\)2. \(x + 7k = 0\)

Step 4: Solve each equation

To find the values of \(x\), solve each equation from the previous step.1. \(x = 0\) is already solved.2. Solve \(x + 7k = 0\) for \(x\):\[ x = -7k \]

Key Concepts

Zero Product Property

The Zero Product Property is an essential principle in algebra that helps solve equations like the one we're working with. It states that if the product of two numbers is zero, then at least one of the numbers must also be zero.
This is crucial because it allows us to break down equations into simpler parts.
For example, when we factor an equation into

• \(a \cdot b = 0 \)

we can directly conclude that either

• \( a = 0 \)
• \( b = 0 \)

This property is utilized widely when solving quadratic equations. In our exercise, once we factored \(x(x + 7k) = 0\), we applied this concept to find the solutions by setting each factor equal to zero. Hence, we get two potential answers: \(x = 0\) and \(x = -7k\). This simple idea allows us to handle even more complex equations by reducing them into manageable pieces.

Common Factor

Finding a common factor is a crucial first step in simplifying and solving equations through factoring. A factor is a term that divides each part of the equation smoothly without leaving any remainders.
When searching for common factors, look for elements shared by every term in the equation.
For instance:

• In \(x^2 + 7kx = 0\), both terms, \(x^2\) and \(7kx\), share the variable \(x\) as a factor.

By factoring out the common \(x\), the equation simplifies to \(x(x + 7k) = 0\). This not only makes the equation simpler to work with but also paves the way to apply the Zero Product Property efficiently. Always start by identifying and factoring out the common factor, as it often reveals the path to solving the equation effectively.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically in the form \(ax^2 + bx + c = 0\). Solving them can seem daunting at first, but they become more manageable with techniques like factoring.
In our specific problem, despite the absence of a constant term, it fits the structure of a quadratic equation because it involves an \(x^2\) term.Quadratics can be solved using various methods:

• **Factoring**: This involves finding two binomials whose product gives the original quadratic, like how we transformed \(x^2 + 7kx = 0\) into \(x(x + 7k) = 0\).
• **Quadratic Formula**: A more advanced method used when factoring isn’t easy, giving solutions directly from the formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
• **Completing the Square**: Another technique where the quadratic is transformed into a perfect square trinomial, then solved accordingly.

In practice, factoring is often the simplest way when a common factor like \(x\) makes the equation much easier to tackle. Mastering these approaches will give you a strong toolkit for solving any quadratic equation you encounter.