Solving The Quadratic Equation: 3x² - 25x + 44x + 12 = 0

Welcome to our deep dive into solving the quadratic equation 3x² - 25x + 44x + 12 = 0! In this article, we'll break down the process step-by-step, making it understandable and engaging for everyone. Whether you're a student grappling with algebra or just curious about how these equations work, you're in the right place. We'll explore the fundamental concepts, demonstrate practical methods for finding the solutions, and discuss the importance of quadratic equations in various fields. Get ready to unlock the secrets of this fascinating mathematical problem!

Understanding Quadratic Equations

Before we jump into solving 3x² - 25x + 44x + 12 = 0, let's get a solid grasp of what quadratic equations are. At its core, a quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in this case, 'x') is 2. The standard form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are coefficients (constants), and 'a' cannot be zero. If 'a' were zero, it would simply become a linear equation. Quadratic equations are fundamental in mathematics because they describe many real-world phenomena, from projectile motion and optimization problems to economic modeling and engineering designs. The graphs of quadratic functions are parabolas, which have a distinctive U-shape, either opening upwards or downwards, depending on the sign of the 'a' coefficient. Understanding these basic principles sets the stage for tackling specific problems like the one we have before us.

It's crucial to recognize the components of our specific equation: 3x² - 25x + 44x + 12 = 0. First, we need to simplify it. Notice that we have two terms with 'x': -25x and +44x. Combining these like terms, we get -25x + 44x = 19x. So, our simplified equation becomes 3x² + 19x + 12 = 0. Now, this fits the standard form ax² + bx + c = 0 perfectly, where:

• a = 3 (the coefficient of x²)
• b = 19 (the coefficient of x)
• c = 12 (the constant term)

Identifying these coefficients is the first critical step in applying any of the standard methods for solving quadratic equations. The values of 'a', 'b', and 'c' will be plugged into formulas and used in various manipulations to isolate the variable 'x' and find its values, which are also known as the roots or solutions of the equation. The nature of these roots (real, complex, distinct, or repeated) depends on the discriminant, which we'll discuss later. For now, let's celebrate this simplification – transforming the initial expression into a clear, standard quadratic equation is a significant milestone in our problem-solving journey.

Methods for Solving Quadratic Equations

There are several tried-and-true methods for solving quadratic equations, and the best one to use often depends on the specific equation and your personal preference. For our equation, 3x² + 19x + 12 = 0, we can consider factoring, completing the square, or using the quadratic formula. Let's explore each of these.

1. Factoring (if possible)

Factoring is often the quickest method if the quadratic expression can be easily factored. The goal is to rewrite the equation in the form (px + q)(rx + s) = 0. To do this, we look for two binomials that multiply together to give us our original quadratic. For 3x² + 19x + 12 = 0, we need to find two numbers that multiply to give a * c (which is 3 * 12 = 36) and add up to b (which is 19). Let's list pairs of factors of 36 and see which pair adds up to 19:

• 1 and 36 (sum = 37)
• 2 and 18 (sum = 20)
• 3 and 12 (sum = 15)
• 4 and 9 (sum = 13)

Hmm, it seems that there aren't two integer factors of 36 that add up to exactly 19. This suggests that our quadratic expression might not be easily factorable using integers. While it's possible it could be factored with fractions or irrational numbers, it's not a straightforward approach in this case. When factoring doesn't immediately present itself, we move on to more robust methods. Don't worry, there are other powerful ways to solve this!

2. Completing the Square

Completing the square is a method that transforms the quadratic equation into a form where you can easily take the square root of both sides. It's particularly useful for deriving the quadratic formula itself. Let's apply it to 3x² + 19x + 12 = 0.

First, we want the coefficient of x² to be 1. So, we divide the entire equation by 3:

x² + (19/3)x + 4 = 0

Next, we move the constant term to the right side of the equation:

x² + (19/3)x = -4

Now, here's the