Solutions To The Quadratic Equation X^2 + 6x - 5 = 0 A Comprehensive Guide

In the realm of mathematics, quadratic equations hold a significant place. These equations, characterized by their second-degree polynomial form, appear in various scientific and engineering applications. Mastering the techniques to solve these equations is a fundamental skill for anyone venturing into advanced mathematical studies. This article delves into the intricacies of solving a specific quadratic equation, providing a step-by-step approach and highlighting key concepts along the way.

Understanding Quadratic Equations

Before we dive into the specifics of the equation at hand, let's establish a solid foundation by defining what a quadratic equation is and exploring its general form. A quadratic equation is a polynomial equation of the second degree. This means that the highest power of the variable (typically represented as 'x') is 2. The general form of a quadratic equation is expressed as:

$ax^2 + bx + c = 0$

where 'a', 'b', and 'c' are constants, and 'a' is not equal to 0. The coefficients 'a', 'b', and 'c' play crucial roles in determining the nature and values of the solutions (also known as roots) of the quadratic equation. The solutions represent the values of 'x' that satisfy the equation, making the expression equal to zero.

There are several methods to solve quadratic equations, each with its own strengths and applicability. The most common methods include:

1. Factoring: This method involves expressing the quadratic expression as a product of two linear factors. Factoring is efficient when the quadratic expression can be easily factored.
2. Completing the Square: This method transforms the quadratic equation into a perfect square trinomial, allowing for direct solution by taking square roots.
3. Quadratic Formula: This formula provides a direct solution for any quadratic equation, regardless of its factorability. It is a versatile method that always yields the solutions, albeit sometimes involving more complex calculations.

Solving the Equation: $x^2 + 6x - 5 = 0$

Now, let's turn our attention to the specific quadratic equation presented: $x^2 + 6x - 5 = 0$. Our goal is to find the values of 'x' that satisfy this equation. We will explore different methods to solve it and arrive at the correct solutions.

1. Attempting Factoring

The first approach we can try is factoring. We need to find two numbers that multiply to -5 (the constant term) and add up to 6 (the coefficient of the 'x' term). However, it quickly becomes apparent that there are no two integers that satisfy these conditions. The factors of -5 are (-1, 5) and (1, -5), and neither pair adds up to 6. Therefore, factoring is not a suitable method for this particular equation.

2. Completing the Square

Since factoring didn't work, let's try the method of completing the square. This method involves manipulating the equation to create a perfect square trinomial on one side. Here's how we apply it to our equation:

1. Move the constant term to the right side: $x^2 + 6x = 5$

2. Take half of the coefficient of the 'x' term (which is 6), square it (3^2 = 9), and add it to both sides of the equation: $x^2 + 6x + 9 = 5 + 9$

3. Rewrite the left side as a squared binomial: $(x + 3)^2 = 14$

4. Take the square root of both sides: $x + 3 = extbf{±}√14$

5. Isolate 'x' by subtracting 3 from both sides: $x = -3 extbf{±} √14$

Thus, using the method of completing the square, we find that the solutions are $x = -3 + √14$ and $x = -3 - √14$.

3. Applying the Quadratic Formula

The quadratic formula provides a universal solution for any quadratic equation. It is derived from the method of completing the square and can be directly applied to find the roots. The quadratic formula is given by:

x = rac{-b extbf{±}√(b^2 - 4ac)}{2a}

where 'a', 'b', and 'c' are the coefficients of the quadratic equation in the form $ax^2 + bx + c = 0$.

For our equation, $x^2 + 6x - 5 = 0$, we have a = 1, b = 6, and c = -5. Substituting these values into the quadratic formula, we get:

x = rac{-6 extbf{±}√(6^2 - 4 * 1 * -5)}{2 * 1}

x = rac{-6 extbf{±}√(36 + 20)}{2}

x = rac{-6 extbf{±}√56}{2}

x = rac{-6 extbf{±}√(4 * 14)}{2}

x = rac{-6 extbf{±}2√14}{2}

$x = -3 extbf{±} √14$

As we can see, the quadratic formula yields the same solutions as the method of completing the square: $x = -3 + √14$ and $x = -3 - √14$.

Analyzing the Solutions

We have successfully found the solutions to the quadratic equation $x^2 + 6x - 5 = 0$ using both completing the square and the quadratic formula. The solutions are $x = -3 + √14$ and $x = -3 - √14$. These are two distinct real roots. The presence of the square root term indicates that the roots are irrational numbers. It is important to note that quadratic equations can have two real roots, one real root (a repeated root), or two complex roots, depending on the discriminant ($b^2 - 4ac$) within the quadratic formula.

Identifying the Correct Answer

Based on our calculations, the solutions to the quadratic equation $x^2 + 6x - 5 = 0$ are $x = -3 extbf{±} √14$. Comparing this result with the options provided:

A. $x = -6 extbf{±} √14$

B. $x = -3 extbf{±} √56$

C. $x = -3 extbf{±} √14$

D. $x = -6 extbf{±} √56$

We can clearly see that option C, $x = -3 extbf{±} √14$, is the correct answer.

Option B can also be a valid answer if we simplify the square root of 56: √56 = √(4 * 14) = 2√14. Dividing both sides of the equation in option B by 2, we get the same result as option C.

Conclusion

Solving quadratic equations is a fundamental skill in algebra. In this article, we explored the equation $x^2 + 6x - 5 = 0$ and demonstrated how to solve it using two methods: completing the square and the quadratic formula. Both methods led us to the same solutions, $x = -3 + √14$ and $x = -3 - √14$. Understanding the different methods and their applications allows for a more comprehensive approach to solving quadratic equations. By mastering these techniques, you can confidently tackle a wide range of mathematical problems and real-world applications involving quadratic relationships.

In summary, the process of solving quadratic equations involves identifying the appropriate method, applying it carefully, and interpreting the results in the context of the problem. Whether it's factoring, completing the square, or using the quadratic formula, each method offers a unique pathway to finding the solutions. The key is to practice and develop a strong understanding of the underlying concepts, enabling you to solve quadratic equations with ease and accuracy.