Problem 17 - College Algebra

Question

Solve the quadratic equation by factoring. $$ 7 x^{2}+3 x=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = -\frac{3}{7}\).

Step 1: Identify the common factor

The equation given is \(7x^2 + 3x = 0\). First, notice that both terms have a common factor of \(x\). Factoring \(x\) from each term gives: \(x(7x + 3) = 0\).

Step 2: Apply the zero-product property

According to the zero-product property, if a product of two factors equals zero, then at least one of the factors must be zero. This means either \(x = 0\) or \(7x + 3 = 0\).

Step 3: Solve for x in each factor

For the first factor, \(x = 0\) is already a solution. For the second factor, solve \(7x + 3 = 0\) by subtracting 3 from both sides to get \(7x = -3\), then divide by 7 to find \(x = -\frac{3}{7}\).

Step 4: State the solutions

The solutions to the equation \(7x^2 + 3x = 0\) are \(x = 0\) and \(x = -\frac{3}{7}\).

Key Concepts

Factoring

Factoring a quadratic equation means to express it as a product of its factors. In this context, factors are expressions that multiply together to form the quadratic equation. When you are given an equation like \(7x^2 + 3x = 0\), the goal is to simplify it into a form that can be easily solved. To factor such an equation, look for common factors among the terms.

Here's how this works:

• Check each term of the equation for any common factors.
• In \(7x^2 + 3x = 0\), both terms share \(x\) as a common factor.
• Factor out \(x\), transforming the equation into \(x(7x + 3) = 0\).

This form makes it easier to apply additional properties, like the zero-product property. Factoring is a critical step in solving quadratic equations as it simplifies what might seem like a complex equation into more manageable parts.

Zero-Product Property

The zero-product property is a fundamental concept in algebra that makes solving factored equations straightforward. According to this property, if the product of two factors equals zero, then at least one factor must be zero.

In our equation \(x(7x + 3) = 0\):

• Apply the zero-product property by setting each factor equal to zero.
• For the first factor \(x\), set \(x = 0\).
• For the second factor \(7x + 3\), set \(7x + 3 = 0\).

By doing this, you effectively split the problem into simpler parts that can be solved independently. For \(x = 0\), the solution is straightforward. For the equation \(7x + 3 = 0\), you would further simplify to find the other solution.

Common Factor

A common factor is a number or variable that divides exactly into each term of an expression. Identifying and factoring out common factors is an essential skill because it simplifies the expressions, making them easier to solve or manipulate.

In our equation example \(7x^2 + 3x = 0\):

• Both terms \(7x^2\) and \(3x\) share \(x\) as a common factor.
• Recognizing this common factor allows you to rewrite the equation as \(x(7x + 3) = 0\).

By factoring out \(x\), the equation becomes much simpler and sets you up perfectly for using the zero-product property. Understanding how to find common factors is helpful in factoring various expressions beyond just quadratic equations.