Solving Quadratic Equations By Factoring A Step-by-Step Guide

Jun 26, 2025 by ADMIN 62 views

Solve the quadratic equation $x^2-x+18=-8x+8$ by factoring.

Solving Quadratic Equations by Factoring: A Comprehensive Guide

In the realm of mathematics, solving quadratic equations is a fundamental skill. Among the various methods available, factoring stands out as an elegant and efficient technique. This article delves into the process of solving quadratic equations by factoring, using the example equation $x^2 - x + 18 = -8x + 8$ as a practical illustration. We will explore the underlying principles, step-by-step procedures, and nuances of this method, providing a comprehensive guide for students and enthusiasts alike.

Understanding Quadratic Equations

Before diving into the specifics of factoring, it's crucial to grasp the essence of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is expressed as:

$ax^2 + bx + c = 0$

where 'a', 'b', and 'c' are constants, and 'a' is not equal to 0. The solutions to a quadratic equation, also known as roots or zeros, are the values of 'x' that satisfy the equation. These roots represent the points where the parabola defined by the quadratic equation intersects the x-axis. Solving quadratic equations allows us to find these crucial points and gain a deeper understanding of the equation's behavior.

The Power of Factoring

Factoring is a technique that involves expressing a quadratic equation as a product of two linear factors. In other words, we aim to rewrite the equation in the form:

$(x + p)(x + q) = 0$

where 'p' and 'q' are constants. The beauty of this form lies in 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. This property allows us to find the solutions by setting each factor equal to zero and solving for 'x'. Factoring is not just a method for finding solutions; it's a powerful tool for simplifying expressions, understanding polynomial behavior, and laying the foundation for more advanced mathematical concepts.

Step-by-Step Guide to Solving by Factoring

Let's now tackle the given equation, $x^2 - x + 18 = -8x + 8$ , using the factoring method. This step-by-step guide will break down the process into manageable steps, ensuring a clear understanding of the technique.

Step 1: Rearrange the Equation

The first step is to rearrange the equation into the standard quadratic form, $ax^2 + bx + c = 0$ . This involves moving all terms to one side of the equation, leaving zero on the other side. To do this, we add $8x$ and subtract $8$ from both sides of the equation:

$x^2 - x + 18 + 8x - 8 = -8x + 8 + 8x - 8$

This simplifies to:

$x^2 + 7x + 10 = 0$

Now, we have the equation in the standard form, ready for the next step.

Step 2: Factor the Quadratic Expression

The core of the factoring method lies in decomposing the quadratic expression into two linear factors. This involves finding two numbers that add up to the coefficient of the 'x' term (which is 7 in this case) and multiply to the constant term (which is 10). Let's call these numbers 'p' and 'q'. We need to find 'p' and 'q' such that:

$p + q = 7$

$p * q = 10$

By carefully considering the factors of 10, we can identify that 2 and 5 satisfy these conditions:

$2 + 5 = 7$

$2 * 5 = 10$

Therefore, we can rewrite the quadratic expression as:

$x^2 + 2x + 5x + 10 = 0$

Now, we can factor by grouping:

$x(x + 2) + 5(x + 2) = 0$

Notice that $(x + 2)$ is a common factor. Factoring it out, we get:

$(x + 2)(x + 5) = 0$

We have successfully factored the quadratic expression into two linear factors. This step is crucial for applying the zero-product property and finding the solutions.

Step 3: Apply the Zero-Product Property

The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. In our case, we have:

$(x + 2)(x + 5) = 0$

This implies that either $(x + 2) = 0$ or $(x + 5) = 0$ . We now have two simple linear equations to solve.

Step 4: Solve for x

Let's solve each linear equation separately:

For $(x + 2) = 0$ , we subtract 2 from both sides to get:

$x = -2$

For $(x + 5) = 0$ , we subtract 5 from both sides to get:

$x = -5$

Therefore, the solutions to the quadratic equation $x^2 + 7x + 10 = 0$ are $x = -2$ and $x = -5$ . These are the values of 'x' that make the equation true.

Verifying the Solutions

To ensure the accuracy of our solutions, it's always a good practice to substitute them back into the original equation and verify that they satisfy the equation. Let's verify our solutions for $x^2 - x + 18 = -8x + 8$ :

For $x = -2$ :

$(-2)^2 - (-2) + 18 = -8(-2) + 8$

$4 + 2 + 18 = 16 + 8$

$24 = 24$

The equation holds true for $x = -2$ .

For $x = -5$ :

$(-5)^2 - (-5) + 18 = -8(-5) + 8$

$25 + 5 + 18 = 40 + 8$

$48 = 48$

The equation also holds true for $x = -5$ . This confirms that our solutions are correct.

Advantages and Limitations of Factoring

Factoring is a powerful method for solving quadratic equations, but it's essential to understand its advantages and limitations. Let's delve into these aspects.

Advantages

Simplicity: Factoring is often the most straightforward method when the quadratic equation can be easily factored. It avoids complex formulas and provides a clear understanding of the roots.
Efficiency: When applicable, factoring is a quick and efficient way to find the solutions. It requires minimal calculations and can be done mentally in many cases.
Conceptual Understanding: Factoring reinforces the understanding of polynomial manipulation and the relationship between factors and roots. It provides a deeper insight into the structure of quadratic equations.

Limitations

Not Always Applicable: Not all quadratic equations can be easily factored using integers. In such cases, other methods like the quadratic formula or completing the square are necessary.
Trial and Error: Factoring often involves some trial and error, especially when the coefficients are large or the factors are not immediately obvious. This can be time-consuming and frustrating for some students.
Limited to Rational Roots: Factoring primarily works well when the roots of the equation are rational numbers. If the roots are irrational or complex, factoring becomes challenging or impossible.

Alternative Methods for Solving Quadratic Equations

While factoring is a valuable technique, it's not the only method for solving quadratic equations. Other methods, such as the quadratic formula and completing the square, provide alternative approaches that are applicable in a wider range of scenarios. Let's briefly explore these methods.

Quadratic Formula

The quadratic formula is a universal method that can be used to solve any quadratic equation, regardless of whether it can be factored or not. The formula is given by:

$x = rac{-b ext{±} ext{√(}b^2 - 4ac)}{2a}$

where 'a', 'b', and 'c' are the coefficients of the quadratic equation in the standard form $ax^2 + bx + c = 0$ . The quadratic formula provides a direct way to calculate the roots, even when they are irrational or complex. This method is particularly useful when factoring is difficult or impossible.

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, which can then be factored as a squared binomial. This method is particularly useful for deriving the quadratic formula and understanding the relationship between the coefficients and the roots. Completing the square can be more involved than factoring or using the quadratic formula, but it provides a deeper understanding of the structure of quadratic equations.

Conclusion

Solving quadratic equations by factoring is a fundamental skill in algebra. This article has provided a comprehensive guide to this method, using the example equation $x^2 - x + 18 = -8x + 8$ to illustrate the step-by-step process. We have explored the underlying principles, advantages, and limitations of factoring, as well as alternative methods for solving quadratic equations. By mastering factoring and other techniques, students can gain a solid foundation in algebra and tackle a wide range of mathematical problems. Remember, practice is key to developing proficiency in solving quadratic equations, so keep practicing and exploring different approaches. The ability to solve quadratic equations is not just a mathematical skill; it's a valuable tool for problem-solving and critical thinking in various fields.