Solving 2x² + 7x + 3 = 0 A Step-by-Step Guide To Finding Solutions

Jul 9, 2025 by Jeany 67 views

Solving Quadratic Equations: A Step-by-Step Guide to Finding the Roots of 2x² + 7x + 3 = 0

In this comprehensive guide, we will delve into the process of solving quadratic equations, specifically focusing on the equation 2x² + 7x + 3 = 0. Quadratic equations, which are polynomial equations of the second degree, play a vital role in various fields, including mathematics, physics, engineering, and computer science. Understanding how to solve them is a fundamental skill for anyone pursuing these disciplines.

Understanding Quadratic Equations

Before we dive into the solution, let's first understand the anatomy of a quadratic equation. A quadratic equation is generally expressed in the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable we aim to solve for. The solutions to a quadratic equation are also known as its roots or zeros.

In our specific case, the equation 2x² + 7x + 3 = 0 fits this form, where a = 2, b = 7, and c = 3. Our goal is to find the values of 'x' that satisfy this equation. There are several methods to solve quadratic equations, and we will explore one of the most common methods: factoring.

Method 1: Solving Quadratic Equations by Factoring

Factoring is a powerful technique for solving quadratic equations, especially when the equation can be easily factored. The idea behind factoring is to rewrite the quadratic expression as a product of two linear expressions. Let's break down the steps involved in factoring the equation 2x² + 7x + 3 = 0.

Step 1: Find two numbers that multiply to ac and add up to b

In our equation, a = 2, b = 7, and c = 3. So, we need to find two numbers that multiply to ac = 2 * 3 = 6 and add up to b = 7. After some thought, we can identify these numbers as 6 and 1, since 6 * 1 = 6 and 6 + 1 = 7.

Step 2: Rewrite the middle term using the two numbers found

Now that we have the numbers 6 and 1, we can rewrite the middle term (7x) in our equation as the sum of 6x and 1x. This gives us the following: 2x² + 6x + 1x + 3 = 0.

Step 3: Factor by grouping

Next, we will factor by grouping. We group the first two terms and the last two terms together: (2x² + 6x) + (1x + 3) = 0. Now, we factor out the greatest common factor (GCF) from each group.

From the first group (2x² + 6x), the GCF is 2x. Factoring out 2x gives us 2x(x + 3). From the second group (1x + 3), the GCF is 1. Factoring out 1 gives us 1(x + 3). So, our equation now looks like this: 2x(x + 3) + 1(x + 3) = 0.

Notice that we now have a common factor of (x + 3) in both terms. We can factor out (x + 3) to get (2x + 1)(x + 3) = 0.

Step 4: Set each factor equal to zero and solve for x

We have now successfully factored the quadratic equation into two linear factors. To find the solutions, we set each factor equal to zero and solve for x:

2x + 1 = 0 Subtracting 1 from both sides gives us 2x = -1. Dividing both sides by 2 gives us x = -1/2.
x + 3 = 0 Subtracting 3 from both sides gives us x = -3.

Therefore, the solutions to the equation 2x² + 7x + 3 = 0 are x = -1/2 and x = -3.

Checking the Solutions

To ensure the accuracy of our solutions, we can substitute them back into the original equation and verify that they satisfy the equation.

Checking x = -1/2

Substituting x = -1/2 into the equation 2x² + 7x + 3 = 0 gives us:

2(-1/2)² + 7(-1/2) + 3 = 2(1/4) - 7/2 + 3 = 1/2 - 7/2 + 3 = -6/2 + 3 = -3 + 3 = 0

Since the equation holds true, x = -1/2 is a valid solution.

Checking x = -3

Substituting x = -3 into the equation 2x² + 7x + 3 = 0 gives us:

2(-3)² + 7(-3) + 3 = 2(9) - 21 + 3 = 18 - 21 + 3 = 0

Since the equation holds true, x = -3 is also a valid solution.

Analyzing the Given Options

Now, let's analyze the given options in the context of our solutions:

A. x = 7: This is not a solution to the equation.

B. x = -3: This is a solution to the equation, as we found through factoring and verification.

C. x = 2: This is not a solution to the equation.

D. x = 3: This is not a solution to the equation.

E. x = 4: This is not a solution to the equation.

F. x = -1/2: This is a solution to the equation, as we found through factoring and verification.

Conclusion: Mastering Quadratic Equations

In conclusion, we have successfully solved the quadratic equation 2x² + 7x + 3 = 0 using the factoring method. We identified the solutions as x = -1/2 and x = -3, and we verified these solutions by substituting them back into the original equation. Understanding how to solve quadratic equations is a crucial skill in mathematics and its applications, and mastering techniques like factoring will greatly enhance your problem-solving abilities. Remember to always check your solutions to ensure accuracy and a deeper understanding of the underlying concepts.

By carefully following the steps outlined in this guide, you can confidently tackle quadratic equations and expand your mathematical toolkit. Whether you are a student, engineer, or simply someone with a passion for mathematics, the ability to solve quadratic equations will undoubtedly prove valuable in your academic and professional pursuits. So, continue practicing and exploring the world of quadratic equations – the journey of mathematical discovery is always rewarding!