Hey there! Have you ever stumbled upon a quadratic equation and wondered, “How do I even solve this?” Don’t worry, you’re not alone. In this blog, we’ll dive into solving a specific quadratic equation, 4x² – 5x – 12 = 0, step by step. By the end, you’ll understand how to crack it and how these equations play a crucial role in real-life problems.
Now, solving quadratic equations might initially sound intimidating, but trust me, it’s easier than you think. Whether you’re solving it for school, work, or just sharpening your math skills, this guide will break it down in plain, conversational language. Let’s go!
Table of Contents: 4x² – 5x – 12 = 0
- What is a Quadratic Equation?
- Breaking Down 4x² – 5x – 12 = 0
- How to Find the Roots of This Equation?
- Analyzing the Roots
- Frequently Asked Questions (FAQs)
- Summary
What is a Quadratic Equation? 4x² – 5x – 12 = 0
Before we jump straight into solving 4x² – 5x – 12 = 0, let’s clarify what a quadratic equation is.
Simply put, a quadratic equation is a second-degree polynomial equation where the highest power of the variable (x) is 2. These equations come in the form:ax² + bx + c = 0Here:
- a is the coefficient of the quadratic term (the x² term).
- b is the coefficient of the linear term (the x term).
- c is the constant (the number without x).
Quadratic equations are everywhere—from physics to computer science to business calculations. Solving them gives you the roots, or the values of x, that make the equation true.
Breaking Down 4x² – 5x – 12 = 0
Now, dissect the specific quadratic equation with 4x² – 5x – 12 = 0.
In this equation:
- 4x² is the quadratic term.
- -5x is the linear term.
- -12 is the constant term.
To solve this equation, we’ll use the quadratic formula. Ready to dive in?
How to Find the Roots of This Equation?
Here’s where things get interesting. We will use the quadratic formula, the go-to tool for solving any quadratic equation. If you don’t remember the formula, no worries. It goes like this:
x = (-b ± √(b² – 4ac)) / (2a)
For our equation, 4x² – 5x – 12 = 0, we already know the values of a, b, and c:
- a = 4
- b = -5
- c = -12
Step-by-Step Breakdown:
- Identify the coefficients:
- a = 4
- b = -5
- c = -12
- Substitute the values into the quadratic formula:
- x = (-(-5) ± √((-5)² – 4 * 4 * (-12))) / (2 * 4)
- Simplify inside the square root:
- First, calculate (-5)² = 25
- Then, calculate 4 * 4 * -12 = -192
- Inside the square root, you get 25 + 192 = 217
- Calculate the two possible roots:
- Root 1 (Positive):
- x = (5 + √217) / 8
- Root 2 (Negative):
- x = (5 – √217) / 8
- Root 1 (Positive):
Analyzing the Roots
Now that we’ve got the formula plugged in, let’s talk about these roots.
Root 1: Positive Root
The first root comes from the positive part of the formula. Here’s the step-by-step simplification:
- x = (5 + √217) / 8
- Since √217 is approximately 14.73, we can approximate the positive root:
- x ≈ (5 + 14.73) / 8
- x ≈ 19.73 / 8
- x ≈ 2.47
So, one of the 4x² – 5x – 12 = 0 roots is approximately x ≈ 2.47.
Root 2: 
For the second root, we use the negative part of the quadratic formula:
- x = (5 – √217) / 8
- Again, with √217 ≈ 14.73, we can calculate:
- x ≈ (5 – 14.73) / 8
- x ≈ -9.73 / 8
- x ≈ -1.22
So, the other root is approximately x ≈ -1.22.
FAQs
Q1: What is the quadratic formula again?
- The quadratic formula is: x = (-b ± √(b² – 4ac)) / (2a). You use it to solve any quadratic equation, whether it’s simple or complex.
Q2: Why do we use the quadratic formula?
- The quadratic formula is a reliable method to find the roots of any quadratic equation, especially when factoring isn’t an option.
Q3: What are the roots of 4x² – 5x – 12 = 0?
- The roots are approximately x ≈ 2.47 and x ≈ -1.22.
Q4: What is the importance of finding the roots of a quadratic equation?
- Finding roots helps in various fields, such as physics, engineering, and economics. In real-life applications, roots can represent time, distance, or even probabilities.
Summary
Solving the quadratic equation 4x² – 5x – 12 = 0 using the quadratic formula is pretty straightforward once you know the steps. We found that the two roots of the equation are approximately x ≈ 2.47 and x ≈ -1.22.
Understanding quadratic equations strengthens your mathematical skills and equips you with problem-solving techniques applicable to real-world situations. Whether it’s calculating trajectories, optimizing business profits, or designing algorithms, quadratic equations are at the core of so many fields.
Next time you face a quadratic equation, don’t sweat it. Break it down, apply the quadratic formula, and solve it step by step.
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