Solving 7x² - 20x = 3 A Step-by-Step Guide

Hey guys! Let's dive into solving the quadratic equation 7x² - 20x = 3. Quadratic equations might seem intimidating at first, but with the right approach, they become quite manageable. This article will walk you through the step-by-step process of solving this particular equation, discussing the different methods available, and giving you a solid understanding of how to tackle similar problems in the future. Whether you're a student brushing up on algebra or just someone who enjoys a good math challenge, you've come to the right place. We’ll break down each step, ensuring that you not only get the answer but also grasp the underlying concepts. So, let's get started and unravel the mystery of this quadratic equation together!

Understanding Quadratic Equations

Before we jump into the specifics of 7x² - 20x = 3, it's crucial to understand what a quadratic equation actually is. 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 ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. If a were zero, the equation would become a linear equation instead of a quadratic one.

The coefficients a, b, and c play significant roles in determining the nature and solutions of the equation. The term ax² is the quadratic term, bx is the linear term, and c is the constant term. Understanding this standard form is the first step in solving any quadratic equation because it helps us identify the coefficients needed for various solution methods.

Now, let’s consider our specific equation, 7x² - 20x = 3. To bring it into the standard form, we need to move the constant term from the right side to the left side of the equation. This involves subtracting 3 from both sides, giving us 7x² - 20x - 3 = 0. Now, we can clearly see that a = 7, b = -20, and c = -3. Identifying these coefficients is essential for applying different methods such as factoring, completing the square, or using the quadratic formula.

Why is understanding quadratic equations so important? Well, they appear in various real-world applications, from physics to engineering to economics. For example, quadratic equations can be used to model the trajectory of a projectile, calculate the dimensions of a rectangular area, or even optimize business decisions. By mastering the techniques to solve these equations, you’re not just learning math; you’re gaining tools that are applicable in numerous fields. So, let's move forward and explore the different methods we can use to solve 7x² - 20x - 3 = 0.

Methods to Solve Quadratic Equations

When it comes to solving quadratic equations like 7x² - 20x - 3 = 0, there are several methods we can use. Each method has its own strengths and is suitable for different types of quadratic equations. The three primary methods are factoring, completing the square, and using the quadratic formula. Let's take a closer look at each of these methods and discuss when each one might be the most appropriate choice.

Factoring

Factoring is often the quickest and simplest method if the quadratic equation can be factored easily. Factoring involves rewriting the quadratic expression as a product of two binomials. For example, if we can rewrite ax² + bx + c as (px + q)(rx + s), where p, q, r, and s are constants, then we can easily find the solutions by setting each factor equal to zero. This method relies on 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.

However, not all quadratic equations can be factored easily, especially if the coefficients are large or if the roots are not rational numbers. In such cases, other methods might be more efficient. For our equation, 7x² - 20x - 3 = 0, we will explore factoring as our first approach to see if we can find integer roots quickly. Factoring is a great method to start with because it can save time if the equation factors neatly. We’ll look for two binomials that multiply to give us the original quadratic equation.

Completing the Square

Completing the square is another powerful method for solving quadratic equations. This method involves transforming the quadratic equation into a perfect square trinomial, which can then be easily solved. The general idea is to manipulate the equation ax² + bx + c = 0 into the form (x + h)² = k, where h and k are constants. Once we have this form, we can take the square root of both sides and solve for x.

Completing the square is particularly useful when the quadratic equation cannot be factored easily or when we need to express the quadratic function in vertex form. However, this method can be a bit more complex and time-consuming than factoring, especially if the coefficient of x² (i.e., a) is not 1. Nevertheless, it’s a valuable technique to have in your problem-solving toolkit. While we might not use completing the square for every equation, understanding how it works provides a deeper insight into the structure of quadratic equations.

Quadratic Formula

The quadratic formula is the most versatile method for solving quadratic equations. It can be used to solve any quadratic equation, regardless of whether it can be factored or not. The quadratic formula is derived from the method of completing the square and provides a direct way to find the solutions. For a quadratic equation in the form ax² + bx + c = 0, the quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

This formula gives us two possible solutions for x, which correspond to the two roots of the quadratic equation. The expression inside the square root, b² - 4ac, is called the discriminant, and it tells us about the nature of the roots. If the discriminant is positive, there are two distinct real roots; if it’s zero, there is exactly one real root (a repeated root); and if it’s negative, there are two complex roots.

The quadratic formula is especially useful when factoring is difficult or impossible, and completing the square is too cumbersome. It's a reliable method that always works, making it an essential tool for solving quadratic equations. For our equation, 7x² - 20x - 3 = 0, we will use the quadratic formula as a primary method to ensure we find the solutions accurately. Now, let's get into the step-by-step solution using these methods.

Solving 7x² - 20x = 3 by Factoring

Let’s begin by attempting to solve the quadratic equation 7x² - 20x = 3 using the factoring method. As we discussed earlier, factoring involves rewriting the quadratic expression as a product of two binomials. To start, we need to rewrite the equation in the standard form, which is ax² + bx + c = 0. Subtracting 3 from both sides of the equation gives us:

7x² - 20x - 3 = 0

Now that we have the equation in standard form, we can proceed with factoring. We need to find two binomials, (px + q) and (rx + s), such that their product is equal to 7x² - 20x - 3. This involves finding the correct coefficients p, q, r, and s.

To factor the quadratic expression, we look for two numbers that multiply to the product of the leading coefficient (7) and the constant term (-3), which is -21, and add up to the middle coefficient (-20). The two numbers that satisfy these conditions are -21 and 1 because (-21) * 1 = -21 and (-21) + 1 = -20.

Next, we rewrite the middle term (-20x) using these two numbers:

7x² - 21x + 1x - 3 = 0

Now, we can factor by grouping. We group the first two terms and the last two terms:

(7x² - 21x) + (1x - 3) = 0

Factor out the greatest common factor (GCF) from each group:

7x(x - 3) + 1(x - 3) = 0

Notice that we now have a common binomial factor, which is (x - 3). We can factor this out:

(7x + 1)(x - 3) = 0

Now that we have factored the quadratic equation, we can use the zero-product property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero and solve for x:

1. 7x + 1 = 0
   • Subtract 1 from both sides: 7x = -1
   • Divide by 7: x = -1/7
2. x - 3 = 0
   • Add 3 to both sides: x = 3

Thus, the solutions to the quadratic equation 7x² - 20x - 3 = 0 are x = -1/7 and x = 3. Factoring allowed us to find these solutions in a straightforward manner. This method is efficient when the quadratic expression can be factored neatly. However, let’s also verify these solutions using the quadratic formula to ensure accuracy.

Solving 7x² - 20x = 3 Using the Quadratic Formula

Now, let's tackle the same quadratic equation, 7x² - 20x = 3, but this time using the quadratic formula. As we discussed earlier, the quadratic formula is a powerful tool that can solve any quadratic equation in the form ax² + bx + c = 0. It's especially useful when factoring is not straightforward or when the roots are not rational numbers. The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

First, we need to rewrite the equation in standard form, which we already did in the factoring section:

7x² - 20x - 3 = 0

From this, we can identify the coefficients: a = 7, b = -20, and c = -3. Now, we plug these values into the quadratic formula:

x = (-(-20) ± √((-20)² - 4 * 7 * (-3))) / (2 * 7)

Simplify the expression step by step:

x = (20 ± √(400 + 84)) / 14

x = (20 ± √484) / 14

Since the square root of 484 is 22, we have:

x = (20 ± 22) / 14

Now, we have two possible solutions, one with the plus sign and one with the minus sign:

1. x = (20 + 22) / 14
   • x = 42 / 14
   • x = 3
2. x = (20 - 22) / 14
   • x = -2 / 14
   • x = -1/7

As you can see, we obtained the same solutions as we did with factoring: x = 3 and x = -1/7. This confirms that our factoring method was correct and also demonstrates the reliability of the quadratic formula. The quadratic formula is particularly valuable because it provides a systematic approach to solving quadratic equations, regardless of their complexity. It's a go-to method when factoring seems difficult or when you want to ensure accuracy.

Comparing the Methods and Choosing the Best Approach

We've now solved the quadratic equation 7x² - 20x = 3 using two different methods: factoring and the quadratic formula. Each method has its advantages and disadvantages, and understanding these can help you choose the best approach for solving similar problems in the future. Let's compare these methods and discuss when each one might be most appropriate.

Factoring: Quick but Not Always Applicable

Factoring is often the quickest method when it works. If you can easily identify the factors of the quadratic expression, you can find the solutions relatively quickly. Factoring relies on the zero-product property, which allows us to set each factor equal to zero and solve for x. In our example, factoring 7x² - 20x - 3 = 0 was quite straightforward once we found the correct factors, leading to solutions of x = 3 and x = -1/7.

However, factoring is not always the best choice. Many quadratic equations are not easily factorable, especially if the coefficients are large or if the roots are irrational or complex numbers. Attempting to factor a non-factorable equation can be time-consuming and frustrating. Therefore, while factoring is a great method to try first, it’s essential to recognize when it might be more efficient to use another method.

Quadratic Formula: Reliable and Versatile

The quadratic formula is the most versatile method for solving quadratic equations. It can be used to solve any quadratic equation, regardless of whether it can be factored or not. The formula, x = (-b ± √(b² - 4ac)) / (2a), provides a direct way to find the solutions by plugging in the coefficients a, b, and c. In our case, using the quadratic formula gave us the same solutions as factoring, confirming its accuracy and reliability.

The main advantage of the quadratic formula is its consistency. It always works, making it a reliable tool in your problem-solving arsenal. However, it can be a bit more time-consuming than factoring, especially if the coefficients are large or if the discriminant (b² - 4ac) requires simplification. Despite this, the quadratic formula is often the best choice when factoring is difficult or when you want to ensure you find all solutions, including any irrational or complex roots.

Choosing the Right Method

So, how do you decide which method to use? Here are some guidelines:

1. Try Factoring First: If the quadratic equation looks like it can be easily factored (small coefficients, integer roots), start with factoring. This can save time if it works.
2. If Factoring Fails, Use the Quadratic Formula: If you spend a few minutes trying to factor and don't see an obvious solution, switch to the quadratic formula. This method will always work and ensure you find the solutions.
3. Consider Completing the Square: Completing the square is useful in specific situations, such as when you need to rewrite the quadratic equation in vertex form or when you are dealing with more advanced algebraic manipulations. However, for solving equations, it's generally less efficient than the quadratic formula.

In summary, for the equation 7x² - 20x = 3, both factoring and the quadratic formula worked well, demonstrating the versatility of these methods. The key is to understand the strengths and weaknesses of each method and choose the one that best fits the problem at hand. By mastering these techniques, you’ll be well-equipped to tackle any quadratic equation that comes your way!

Real-World Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have numerous real-world applications across various fields. Understanding these applications can help you appreciate the practical significance of mastering quadratic equations. From physics and engineering to economics and computer science, quadratic equations play a crucial role in modeling and solving real-world problems. Let's explore some key areas where quadratic equations are used extensively.

Physics

In physics, quadratic equations are fundamental in describing the motion of objects under constant acceleration. For example, the trajectory of a projectile, such as a ball thrown into the air, can be modeled using a quadratic equation. The equation relates the object's position to time, taking into account the initial velocity, launch angle, and gravitational acceleration. By solving the quadratic equation, we can determine important parameters such as the maximum height reached by the projectile, the time it takes to hit the ground, and the range of the projectile.

Another application in physics is in the analysis of energy and motion. The kinetic energy of an object, which is the energy it possesses due to its motion, is given by the formula KE = (1/2)mv², where m is the mass and v is the velocity. This equation is quadratic in terms of velocity. Similarly, potential energy in systems like springs can be described using quadratic equations. Understanding these relationships is crucial for solving problems in mechanics and dynamics.

Engineering

Engineers use quadratic equations in a wide range of applications, from designing bridges and buildings to analyzing electrical circuits. In structural engineering, quadratic equations are used to calculate the stresses and strains in beams and other structural elements. For instance, the bending moment in a beam can be related to the applied load through a quadratic relationship. By solving these equations, engineers can ensure the structural integrity and safety of their designs.

In electrical engineering, quadratic equations are used to analyze circuits containing resistors, capacitors, and inductors. The impedance of a circuit element, which is the opposition to the flow of alternating current, can be a quadratic function of frequency. Solving quadratic equations allows engineers to determine the resonant frequencies and optimize circuit performance. Furthermore, quadratic equations are used in control systems engineering to model and control dynamic systems, such as robots and automated machinery.

Economics

In economics, quadratic equations are used to model various relationships, such as cost, revenue, and profit functions. For example, the cost of producing a certain number of units can sometimes be modeled as a quadratic function of the quantity produced. Similarly, the revenue generated from selling a product can be related to the price and quantity sold through a quadratic equation. By analyzing these equations, economists can determine the optimal production levels, pricing strategies, and profit-maximizing points.

Another application is in the study of supply and demand. The equilibrium price and quantity in a market can be found by solving a system of equations, where at least one of the equations might be quadratic. These models help economists understand market dynamics and make predictions about future trends. Quadratic equations also appear in financial models, such as those used to calculate compound interest and investment returns.

Computer Science

In computer science, quadratic equations are used in algorithm design and analysis. For example, the time complexity of certain algorithms, which is a measure of how the runtime of the algorithm grows with the input size, can be expressed using quadratic functions. Algorithms with quadratic time complexity, such as bubble sort and insertion sort, have runtimes that grow proportionally to the square of the input size. Understanding these complexities is crucial for designing efficient algorithms.

Quadratic equations are also used in computer graphics and image processing. For instance, Bézier curves, which are used to create smooth curves and surfaces, are defined using quadratic and cubic polynomials. These curves are essential for rendering graphics in video games, animations, and computer-aided design (CAD) software. Furthermore, quadratic equations are used in image compression techniques and pattern recognition algorithms.

Other Applications

Beyond these fields, quadratic equations have applications in various other areas:

• Astronomy: Calculating the orbits of planets and satellites.
• Environmental Science: Modeling population growth and decay.
• Sports: Analyzing the trajectory of a ball in sports like basketball and baseball.
• Architecture: Designing parabolic arches and other structures.

As you can see, quadratic equations are a powerful tool with widespread applications. By mastering the techniques to solve them, you gain a valuable skill that is relevant across many disciplines. Whether you're a student, engineer, scientist, or simply someone who enjoys problem-solving, understanding quadratic equations will open doors to a deeper understanding of the world around you.

Conclusion

Alright, guys! We've journeyed through the world of quadratic equations, specifically focusing on solving 7x² - 20x = 3. We started by understanding what quadratic equations are and their standard form. Then, we explored various methods to solve them, including factoring and the quadratic formula. We even took a peek at how these equations pop up in real-world scenarios, from physics to economics.

We saw that factoring can be a quick and efficient method when the equation is easily factorable. However, the quadratic formula is the reliable workhorse that always gets the job done, no matter how complex the equation might seem. Choosing the right method depends on the specific equation and your comfort level with each technique. The key takeaway here is that practice makes perfect. The more you work with quadratic equations, the better you'll become at recognizing patterns and choosing the most efficient solution method.

Remember, mastering quadratic equations is not just about getting the right answer; it’s about developing problem-solving skills that can be applied in various fields. Whether you're calculating projectile motion, designing a bridge, or analyzing economic trends, the ability to solve quadratic equations is a valuable asset. So, keep practicing, keep exploring, and don't shy away from a good math challenge. You've got this!