Solving $8x^2 + 7x = 0$ By Factoring: A Step-by-Step Guide

Dec 2, 2025 by Alex Johnson 59 views

Solving the Quadratic Equation $8x^2 + 7x = 0$ by Factoring

Let's dive into solving the quadratic equation $8x^2 + 7x = 0$ using the factoring method. Factoring is a powerful technique to find the roots (or solutions) of a quadratic equation, and in this comprehensive guide, we will break down each step to ensure you understand the process thoroughly. We'll also make sure to express our final answer in reduced fraction form, just in case it's necessary. So, let's get started!

Understanding Quadratic Equations and Factoring

First off, what exactly is a quadratic equation? A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually x) is 2. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where a, b, and c are constants, and a is not equal to 0. In our case, the equation is $8x^2 + 7x = 0$ . Here, a = 8, b = 7, and c = 0.

Factoring is the process of breaking down an expression (in our case, a quadratic expression) into a product of simpler expressions (factors). The goal is to rewrite the quadratic equation in the form $(px + q)(rx + s) = 0$ , where p, q, r, and s are constants. Once we have the equation in this form, we can use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero.

Step-by-Step Solution

1. Identify the Common Factor

In our equation, $8x^2 + 7x = 0$ , notice that both terms have a common factor of x. This makes our job much easier! We can factor out x from both terms:

$x(8x + 7) = 0$

2. Apply the Zero-Product Property

Now that we've factored the equation, we have two factors: x and $(8x + 7)$ . According to the zero-product property, either x = 0 or $(8x + 7) = 0$ . This gives us two separate equations to solve:

Equation 1: $x = 0$
Equation 2: $8x + 7 = 0$

3. Solve Each Equation

The first equation, $x = 0$ , is already solved. This gives us our first solution.

For the second equation, $8x + 7 = 0$ , we need to isolate x. Let's subtract 7 from both sides:

$8x = -7$

Now, divide both sides by 8:

$x = -\frac{7}{8}$

This gives us our second solution.

4. Express the Solution in Reduced Fraction Form

Our solutions are $x = 0$ and $x = - rac{7}{8}$ . The fraction $- rac{7}{8}$ is already in its simplest form, as 7 and 8 have no common factors other than 1. Therefore, we don't need to reduce it further.

The Solutions

So, the solutions to the quadratic equation $8x^2 + 7x = 0$ are:

$x = 0$
$x = - rac{7}{8}$

These are the values of x that make the equation true. You can verify these solutions by substituting them back into the original equation and checking if the equation holds.

Verification

Let's verify our solutions to ensure they are correct.

Verification for $x = 0$

Substitute $x = 0$ into the original equation:

$8(0)^2 + 7(0) = 0$

$8(0) + 0 = 0$

$0 + 0 = 0$

$0 = 0$

The equation holds true for $x = 0$ .

Verification for $x = - rac{7}{8}$

Substitute $x = - rac{7}{8}$ into the original equation:

$8(-\frac{7}{8})^2 + 7(-\frac{7}{8}) = 0$

$8(\frac{49}{64}) - \frac{49}{8} = 0$

$\frac{49}{8} - \frac{49}{8} = 0$

$0 = 0$

The equation also holds true for $x = - rac{7}{8}$ .

Tips for Factoring Quadratic Equations

Factoring quadratic equations might seem daunting at first, but with practice, it becomes much easier. Here are a few tips to keep in mind:

Look for Common Factors: Always start by looking for common factors in all terms of the equation. Factoring out the greatest common factor (GCF) simplifies the equation and makes it easier to handle.
Recognize Special Patterns: Be on the lookout for special patterns like the difference of squares ( $a^2 - b^2 = (a + b)(a - b)$ ) or perfect square trinomials ( $a^2 + 2ab + b^2 = (a + b)^2$ or $a^2 - 2ab + b^2 = (a - b)^2$ ). Recognizing these patterns can significantly speed up the factoring process.
Use the AC Method: For quadratic equations in the form $ax^2 + bx + c = 0$ , the AC method can be very helpful. Multiply a and c, then find two numbers that multiply to this product and add up to b. Use these numbers to split the middle term and factor by grouping.
Practice Regularly: The more you practice factoring, the better you'll become at it. Try different types of quadratic equations and challenge yourself with harder problems.

Common Mistakes to Avoid

When factoring quadratic equations, it's easy to make mistakes. Here are some common pitfalls to watch out for:

Forgetting to Factor Completely: Make sure you factor the equation completely. Sometimes, after factoring once, you might find that one of the factors can be factored further.
Incorrectly Applying the Zero-Product Property: The zero-product property only applies when the equation is set equal to zero. Don't try to apply it if the equation is equal to a non-zero number.
Sign Errors: Pay close attention to the signs of the terms when factoring. A simple sign error can lead to incorrect solutions.
Dividing by a Variable: Avoid dividing both sides of the equation by a variable, as this might cause you to lose a solution (in our case, we could have missed x = 0 if we divided by x).

Alternative Methods for Solving Quadratic Equations

While factoring is a powerful method, it's not always the most efficient or straightforward approach. There are other methods for solving quadratic equations, including:

The Quadratic Formula: The quadratic formula is a universal method that works for any quadratic equation. It's given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

You simply plug in the values of a, b, and c from the equation $ax^2 + bx + c = 0$ and solve for x.
Completing the Square: Completing the square is another method that can be used to solve any quadratic equation. It involves manipulating the equation to create a perfect square trinomial on one side.

Conclusion

In this guide, we've thoroughly explored how to solve the quadratic equation $8x^2 + 7x = 0$ by factoring. We broke down the process into manageable steps, from identifying the common factor to applying the zero-product property and expressing the solutions in reduced fraction form. We also verified our solutions and discussed tips for factoring, common mistakes to avoid, and alternative methods for solving quadratic equations.

Factoring is a fundamental skill in algebra, and mastering it will undoubtedly help you in more advanced math courses. Remember to practice regularly, and don't hesitate to explore other methods if factoring doesn't seem to work. With enough practice, you'll become confident in solving quadratic equations of all kinds.

To further enhance your understanding of quadratic equations and factoring, consider exploring resources like Khan Academy's Quadratic Equations Section. Happy solving!