Solving For Y In Quadratic Equation Y^2 + 4y + 4 = 7 A Step-by-Step Guide

Introduction

In this article, we will delve into the process of finding the value(s) of y in the given quadratic equation: y^2 + 4y + 4 = 7. Quadratic equations are a fundamental concept in algebra, and mastering the techniques to solve them is crucial for various mathematical and scientific applications. We will explore the steps involved in solving this equation, including simplifying it, applying relevant algebraic methods, and interpreting the results. Understanding quadratic equations not only strengthens your mathematical foundation but also enhances your problem-solving skills, which are invaluable in many real-world scenarios. Whether you are a student learning algebra or someone looking to refresh your knowledge, this article provides a comprehensive guide to tackling quadratic equations effectively.

Understanding Quadratic Equations

Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable is two. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable. The solutions to a quadratic equation are also known as its roots or zeros, which represent the values of x that satisfy the equation. These roots can be real or complex numbers, and a quadratic equation can have up to two distinct roots. The quadratic formula, factoring, and completing the square are common methods used to find these roots. Understanding the nature of the roots and the different methods to find them is essential for solving quadratic equations and applying them in various mathematical and scientific contexts. Recognizing the structure and properties of quadratic equations helps in simplifying complex problems and finding accurate solutions. This knowledge is also foundational for understanding higher-degree polynomials and other advanced mathematical concepts.

Step-by-Step Solution

1. Simplify the Equation

The first step in solving the quadratic equation y^2 + 4y + 4 = 7 is to simplify it by moving all terms to one side of the equation, setting it equal to zero. This standard form allows us to easily apply various methods for solving quadratic equations. Subtracting 7 from both sides of the equation gives us: y^2 + 4y + 4 - 7 = 0. Simplifying further, we combine the constant terms: y^2 + 4y - 3 = 0. Now, the equation is in the standard quadratic form ax^2 + bx + c = 0, where a = 1, b = 4, and c = -3. This form is crucial for applying methods such as factoring, completing the square, or using the quadratic formula to find the values of y that satisfy the equation. Ensuring the equation is in this simplified form is a critical step in solving quadratic equations accurately.

2. Apply the Quadratic Formula

To solve the quadratic equation y^2 + 4y - 3 = 0, we can use the quadratic formula, a powerful tool that provides a direct method for finding the roots of any quadratic equation. The quadratic formula is given by: y = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0. In our equation, a = 1, b = 4, and c = -3. Substituting these values into the quadratic formula, we get: y = (-4 ± √(4^2 - 4(1)(-3))) / (2(1)). This substitution sets the stage for calculating the discriminant and finding the two possible values of y that satisfy the equation. The quadratic formula is particularly useful when factoring is not straightforward, making it an indispensable tool for solving quadratic equations.

3. Calculate the Discriminant

The discriminant, represented by the expression b^2 - 4ac, plays a crucial role in determining the nature of the roots of a quadratic equation. It provides valuable information about whether the roots are real, distinct, or complex. In our equation, the discriminant is calculated as follows: 4^2 - 4(1)(-3) = 16 + 12 = 28. Since the discriminant (28) is positive, this indicates that the quadratic equation has two distinct real roots. This information is vital because it tells us that we will find two different real values for y that satisfy the equation. If the discriminant were zero, there would be one real root (a repeated root), and if it were negative, there would be two complex roots. Thus, calculating the discriminant is a key step in understanding the nature of the solutions to a quadratic equation.

4. Determine the Values of y

Now that we have calculated the discriminant and determined that there are two distinct real roots, we can proceed to find the values of y using the quadratic formula. Recall the formula: y = (-b ± √(b^2 - 4ac)) / (2a). Substituting the values a = 1, b = 4, c = -3, and the discriminant 28 into the formula, we get: y = (-4 ± √28) / 2. To simplify further, we can simplify the square root of 28. Since 28 = 4 × 7, √28 = √(4 × 7) = 2√7. Thus, the equation becomes: y = (-4 ± 2√7) / 2. Dividing both terms in the numerator by 2, we obtain two possible values for y: y = -2 + √7 and y = -2 - √7. These are the two distinct real roots of the quadratic equation.

Final Answer

The values of y that satisfy the quadratic equation y^2 + 4y + 4 = 7 are y = -2 + √7 and y = -2 - √7. These solutions represent the points where the parabola defined by the quadratic equation intersects the x-axis. To verify these solutions, we can substitute each value back into the original equation and check if it holds true. This final step ensures the accuracy of our calculations and reinforces our understanding of the solution. Quadratic equations often have two solutions, as seen in this case, reflecting the parabolic nature of the equation. Mastering the techniques to find these solutions is crucial for various applications in mathematics and science, making it a fundamental skill for problem-solving.

Therefore, the correct answer is not explicitly listed among the given options. The derived values are y = -2 + √7 and y = -2 - √7.

Conclusion

In this article, we have thoroughly explored the process of solving the quadratic equation y^2 + 4y + 4 = 7. We began by simplifying the equation to its standard form, which is essential for applying the quadratic formula. The quadratic formula, y = (-b ± √(b^2 - 4ac)) / (2a), is a powerful tool that allows us to find the roots of any quadratic equation. We then calculated the discriminant, b^2 - 4ac, which provided valuable information about the nature of the roots, indicating that our equation had two distinct real solutions. By substituting the values of a, b, and c into the quadratic formula and simplifying, we determined the values of y to be y = -2 + √7 and y = -2 - √7. These solutions represent the values of y that satisfy the equation and can be verified by substituting them back into the original equation.

Understanding how to solve quadratic equations is a fundamental skill in algebra and has numerous applications in various fields, including physics, engineering, and economics. This step-by-step guide not only provides a clear method for solving this particular equation but also enhances your overall problem-solving abilities. Whether you are a student or someone looking to refresh your mathematical skills, mastering these techniques will undoubtedly prove beneficial. The ability to confidently solve quadratic equations opens doors to more advanced mathematical concepts and real-world applications. Therefore, practice and a thorough understanding of the methods discussed are key to success in this area.