Factoring $4x^2 + 9x = -2$: Find X-intercepts

Nov 13, 2025 by Alex Johnson 46 views

Factoring Quadratic Equations: A Comprehensive Guide to Solving $4x^2 + 9x = -2$

Welcome! In this detailed guide, we will walk through the process of factoring the quadratic equation $4x^2 + 9x = -2$ . We'll break down each step, making it easy to understand how to find the factored form and, most importantly, the x-intercepts. Whether you're a student tackling algebra or just refreshing your math skills, this guide is for you. Let's dive in and conquer this equation together!

Understanding Quadratic Equations

Before we tackle our specific equation, let's quickly review what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The standard form of a quadratic equation is $ax^2 + bx + c = 0$ , where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. These equations pop up in various fields, from physics to engineering, making understanding how to solve them crucial.

Now, why is it so important to find the x-intercepts? The x-intercepts are the points where the parabola (the graph of a quadratic equation) crosses the x-axis. These points are also known as the roots or solutions of the equation, and they provide valuable information about the behavior of the quadratic function. Finding these intercepts often involves factoring, which is the method we'll focus on here. Factoring simplifies the equation into a product of binomials, making it easier to identify the values of x that make the equation equal to zero. This is not just a mathematical exercise; understanding x-intercepts helps in real-world applications, such as modeling projectile motion or optimizing designs.

Key Concepts: Factoring and x-intercepts

Factoring: Breaking down a quadratic expression into a product of two binomials.
x-intercepts: The points where the graph of the quadratic equation intersects the x-axis (where y = 0).

Step-by-Step Solution for $4x^2 + 9x = -2$

Let’s get started on solving our equation, $4x^2 + 9x = -2$ . We'll take a step-by-step approach to ensure clarity and understanding. The process involves rearranging the equation into standard form, factoring the quadratic expression, and then finding the x-intercepts.

Step 1: Convert to Standard Form

The first thing we need to do is rewrite the equation in the standard form, $ax^2 + bx + c = 0$ . This involves moving all terms to one side of the equation, leaving zero on the other side. Our equation is $4x^2 + 9x = -2$ . To get it into standard form, we need to add 2 to both sides:

$4x^2 + 9x + 2 = 0$

Now our equation is in the familiar standard form, ready for the next step. This step is crucial because factoring techniques are designed for equations in this format. By ensuring we have a clear zero on one side, we set the stage for easily identifying the solutions once we factor the quadratic expression.

Step 2: Factoring the Quadratic Expression

Next, we need to factor the quadratic expression $4x^2 + 9x + 2$ . Factoring involves breaking the quadratic expression into two binomials. For this, we'll use the factoring by grouping method. This method is particularly useful when the coefficient of $x^2$ is not 1. The goal is to find two numbers that multiply to the product of the leading coefficient (4) and the constant term (2), which is 8, and add up to the middle coefficient (9). These numbers will help us split the middle term and factor by grouping.

Think of two numbers that multiply to 8 and add to 9. The numbers are 8 and 1. Now we split the middle term (9x) using these numbers:

$4x^2 + 8x + 1x + 2 = 0$

Next, we group the terms in pairs and factor out the greatest common factor (GCF) from each pair:

$4x(x + 2) + 1(x + 2) = 0$

Notice that both terms now have a common factor of $(x + 2)$ . We factor this out:

$(4x + 1)(x + 2) = 0$

We have successfully factored the quadratic expression! This step is essential because it transforms our equation into a form where we can easily find the solutions.

Step 3: Finding the x-intercepts

Now that we have the factored form of the equation, $(4x + 1)(x + 2) = 0$ , we can find the x-intercepts. Remember, the x-intercepts are the points where the graph of the equation crosses the x-axis, which means $y = 0$ . To find these points, we set each factor equal to zero and solve for x. This is based on the principle that if the product of two factors is zero, then at least one of the factors must be zero.

So, we have two equations to solve:

$4x + 1 = 0$
$x + 2 = 0$

Let's solve the first equation, $4x + 1 = 0$ :

Subtract 1 from both sides:

$4x = -1$

Divide by 4:

$x = -\frac{1}{4}$

Now, let's solve the second equation, $x + 2 = 0$ :

Subtract 2 from both sides:

$x = -2$

So, the x-intercepts are $x = - rac{1}{4}$ and $x = -2$ . These are the solutions to our original quadratic equation. The x-intercepts tell us where the parabola intersects the x-axis, providing key points for graphing and understanding the function.

Summary of the Solution

Let's recap the steps we took to solve the equation $4x^2 + 9x = -2$ :

Convert to Standard Form: We added 2 to both sides to get $4x^2 + 9x + 2 = 0$ .
Factor the Quadratic Expression: We factored the expression into $(4x + 1)(x + 2) = 0$ .
Find the x-intercepts: We set each factor to zero and solved for x, finding $x = - rac{1}{4}$ and $x = -2$ .

Therefore, the factored form of the equation is $(4x + 1)(x + 2) = 0$ , and the x-intercepts are $x = - rac{1}{4}$ and $x = -2$ .

Graphing the Quadratic Equation

To visualize our solution, let's briefly discuss how the x-intercepts relate to the graph of the quadratic equation. The graph of a quadratic equation is a parabola, a U-shaped curve. The x-intercepts are the points where this curve crosses the x-axis. Knowing the x-intercepts helps us sketch the graph accurately.

In our case, the parabola crosses the x-axis at $x = - rac{1}{4}$ and $x = -2$ . Additionally, we can find the vertex (the highest or lowest point of the parabola) to further refine our graph. The x-coordinate of the vertex can be found using the formula $x = - rac{b}{2a}$ , where a and b are the coefficients from the standard form of the equation. For our equation $4x^2 + 9x + 2 = 0$ , a = 4 and b = 9, so the x-coordinate of the vertex is:

$x = -\frac{9}{2 * 4} = -\frac{9}{8}$

To find the y-coordinate of the vertex, we plug this x-value back into the equation:

$y = 4(-\frac{9}{8})^2 + 9(-\frac{9}{8}) + 2$ $y = 4(\frac{81}{64}) - \frac{81}{8} + 2$ $y = \frac{81}{16} - \frac{162}{16} + \frac{32}{16}$ $y = -\frac{49}{16}$

So, the vertex is at $(-\frac{9}{8}, -\frac{49}{16})$ . With the x-intercepts and the vertex, we can sketch a fairly accurate graph of the parabola. Graphing the equation provides a visual confirmation of our algebraic solution and gives us a better understanding of the quadratic function.

Practical Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have numerous real-world applications. Understanding how to solve them can be incredibly useful in various fields. Let's look at a few examples:

Physics: Quadratic equations are used to model projectile motion. For example, the height of a ball thrown into the air can be described by a quadratic equation. The x-intercepts of the equation can tell us when the ball will hit the ground.
Engineering: Engineers use quadratic equations to design structures, such as bridges and buildings. They need to calculate stress and strain, which often involve quadratic relationships.
Economics: Quadratic functions can model cost, revenue, and profit. Businesses use these models to optimize their operations and maximize profits.
Computer Graphics: Quadratic equations are used in computer graphics to create curves and surfaces. They are essential for rendering images and animations.

By mastering quadratic equations, you gain a powerful tool for solving problems in a wide range of disciplines. The ability to find x-intercepts, factor equations, and understand the graphical representation of quadratic functions is invaluable.

Common Mistakes to Avoid

When working with quadratic equations, it’s easy to make mistakes. Here are some common pitfalls and how to avoid them:

Incorrectly Factoring: Make sure to double-check your factoring. A common mistake is to misidentify the numbers that multiply to the required product and add up to the correct sum. Always expand your factored form to verify it matches the original quadratic expression.
Forgetting to Set Each Factor to Zero: When finding x-intercepts, remember to set each factor equal to zero. It's a common mistake to solve for one factor but forget the other.
Sign Errors: Pay close attention to signs when rearranging equations and factoring. A small sign error can lead to incorrect solutions.
Not Converting to Standard Form: Before factoring, ensure the equation is in standard form ( $ax^2 + bx + c = 0$ ). Trying to factor an equation that isn't in this form can lead to errors.

By being aware of these common mistakes, you can improve your accuracy and confidence in solving quadratic equations. Careful practice and attention to detail are key to mastering this skill.

Conclusion

We've journeyed through the process of factoring the quadratic equation $4x^2 + 9x = -2$ , found its factored form, and identified its x-intercepts. We've also explored the importance of quadratic equations and their applications in various fields. By understanding the steps involved in solving these equations, you're well-equipped to tackle similar problems and appreciate their practical significance.

Remember, the key to mastering quadratic equations is practice. Work through different examples, review the concepts, and don't hesitate to seek help when needed. With dedication and effort, you can become proficient in solving these equations and unlock their power in real-world applications.

For further learning and practice, consider exploring resources like Khan Academy's Quadratic Equations section to deepen your understanding and skills.