Solving The Quadratic Equation X² + 9x - 15 = 0
In this article, we will delve into the process of solving the quadratic equation x² + 9x - 15 = 0. Quadratic equations, which take the general form of ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0, are fundamental concepts in algebra. They appear in various mathematical and real-world applications, ranging from physics and engineering to economics and computer science. Mastering the techniques to solve these equations is essential for anyone seeking a strong foundation in mathematics.
To solve the given quadratic equation, we will explore two primary methods: the quadratic formula and completing the square. The quadratic formula provides a direct solution for any quadratic equation, regardless of its coefficients. It is a versatile tool that guarantees a solution, even when factoring or other simpler methods are not readily applicable. On the other hand, completing the square is a technique that transforms the quadratic equation into a perfect square trinomial, allowing us to isolate the variable and find the solutions. This method is particularly useful for understanding the structure of quadratic equations and their relationship to parabolas.
Before we dive into the methods, it's worth noting that a quadratic equation can have two, one, or no real solutions, depending on the discriminant (b² - 4ac). If the discriminant is positive, there are two distinct real solutions; if it's zero, there is exactly one real solution (a repeated root); and if it's negative, there are no real solutions, but there are two complex solutions. Understanding the nature of the solutions beforehand can help us interpret the results we obtain and verify their validity.
In the following sections, we will meticulously apply both the quadratic formula and the method of completing the square to solve the equation x² + 9x - 15 = 0. We will break down each step, providing explanations and insights to ensure clarity and comprehension. By the end of this article, you will have a thorough understanding of how to solve quadratic equations using these two powerful techniques, and you will be equipped to tackle similar problems with confidence.
The quadratic formula is a powerful tool for solving quadratic equations of the form ax² + bx + c = 0. The formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
This formula provides the solutions for x in terms of the coefficients a, b, and c. It is derived from the method of completing the square and can be applied to any quadratic equation, regardless of whether it can be easily factored. The term b² - 4ac under the square root is known as the discriminant, which we briefly mentioned earlier. The discriminant determines the nature of the roots: if it's positive, there are two distinct real roots; if it's zero, there is one real root (a repeated root); and if it's negative, there are no real roots (but there are two complex roots).
Now, let's apply the quadratic formula to solve the equation x² + 9x - 15 = 0. First, we need to identify the coefficients a, b, and c. In this case:
- a = 1
- b = 9
- c = -15
Next, we substitute these values into the quadratic formula:
x = (-9 ± √(9² - 4 * 1 * -15)) / (2 * 1)
Now, we simplify the expression step by step:
- Calculate the discriminant: 9² - 4 * 1 * -15 = 81 + 60 = 141
- Substitute the discriminant back into the formula: x = (-9 ± √141) / 2
Since 141 is not a perfect square, we leave the square root as it is. Thus, we have two solutions for x:
- x₁ = (-9 + √141) / 2
- x₂ = (-9 - √141) / 2
These are the exact solutions to the quadratic equation. We can approximate these solutions using a calculator:
- x₁ ≈ (-9 + 11.87) / 2 ≈ 1.435
- x₂ ≈ (-9 - 11.87) / 2 ≈ -10.435
Therefore, the solutions to the quadratic equation x² + 9x - 15 = 0, obtained using the quadratic formula, are approximately 1.435 and -10.435. The quadratic formula not only provides the solutions but also demonstrates the power of a general formula in solving mathematical problems. It's a testament to the elegance and efficiency of algebraic methods.
Completing the square is another method for solving quadratic equations. This technique involves transforming the quadratic expression into a perfect square trinomial, which can then be easily solved by taking the square root. While the quadratic formula provides a direct solution, completing the square offers a deeper understanding of the structure of quadratic equations and their relationship to parabolas. It's particularly useful for understanding how the vertex form of a quadratic equation is derived.
To solve the equation x² + 9x - 15 = 0 by completing the square, we follow these steps:
-
Move the constant term to the right side of the equation:
x² + 9x = 15
This step isolates the terms with x on one side, preparing us to complete the square.
-
Complete the square on the left side:
To complete the square, we need to add a constant to both sides of the equation. This constant is calculated by taking half of the coefficient of the x term, squaring it, and adding it to both sides. In this case, the coefficient of the x term is 9. Half of 9 is 9/2, and squaring it gives (9/2)² = 81/4.
x² + 9x + (9/2)² = 15 + (9/2)²
x² + 9x + 81/4 = 15 + 81/4
Now, the left side is a perfect square trinomial, which can be factored as (x + 9/2)².
-
Simplify the equation:
First, convert 15 to a fraction with a denominator of 4: 15 = 60/4.
(x + 9/2)² = 60/4 + 81/4
(x + 9/2)² = 141/4
-
Take the square root of both sides:
√(x + 9/2)² = ±√(141/4)
x + 9/2 = ±√141 / 2
-
Isolate x:
x = -9/2 ± √141 / 2
This gives us two solutions for x:
- x₁ = (-9 + √141) / 2
- x₂ = (-9 - √141) / 2
These are the same solutions we obtained using the quadratic formula. As before, we can approximate these solutions using a calculator:
- x₁ ≈ (-9 + 11.87) / 2 ≈ 1.435
- x₂ ≈ (-9 - 11.87) / 2 ≈ -10.435
Thus, completing the square provides an alternative method to solve the quadratic equation x² + 9x - 15 = 0, yielding the same solutions as the quadratic formula. This technique is particularly valuable for understanding the geometric interpretation of quadratic equations and their relationship to parabolas. It also forms the basis for deriving the quadratic formula itself, highlighting the interconnectedness of algebraic concepts.
Both the quadratic formula and completing the square are powerful methods for solving quadratic equations, but they have different strengths and weaknesses. Understanding these differences can help you choose the most appropriate method for a given problem.
The quadratic formula is a direct and versatile method that can be applied to any quadratic equation. It provides a straightforward solution by substituting the coefficients of the equation into a pre-defined formula. This method is particularly useful when the quadratic equation cannot be easily factored or when the coefficients are complex or irrational numbers. The quadratic formula guarantees a solution, whether it be real or complex, making it a reliable tool for solving a wide range of quadratic equations.
On the other hand, completing the square is a more involved method that requires transforming the quadratic equation into a perfect square trinomial. While it may seem more complex at first, completing the square provides a deeper understanding of the structure of quadratic equations. It reveals the relationship between the equation and its graph, a parabola. Completing the square is also the method used to derive the quadratic formula, highlighting its fundamental importance in the theory of quadratic equations.
When choosing between the two methods, consider the specific equation and your goals. If you need a quick and direct solution, the quadratic formula is often the best choice. It's a plug-and-play method that minimizes the risk of errors. However, if you want to understand the underlying structure of the equation and its geometric interpretation, completing the square is a valuable exercise. It can also be useful for transforming quadratic equations into vertex form, which reveals the vertex and axis of symmetry of the corresponding parabola.
In the case of the equation x² + 9x - 15 = 0, both methods lead to the same solutions, approximately 1.435 and -10.435. However, the process of completing the square provides additional insights into the equation's structure. For instance, it shows how the equation can be rewritten in vertex form, which can be useful for graphing the corresponding parabola.
Ultimately, the choice between the quadratic formula and completing the square depends on your preferences and the specific problem at hand. Mastering both methods will equip you with a comprehensive toolkit for solving quadratic equations and understanding their properties. Each method offers a unique perspective on the nature of quadratic equations, and together, they provide a powerful foundation for further studies in algebra and calculus.
In this article, we have explored two methods for solving the quadratic equation x² + 9x - 15 = 0: the quadratic formula and completing the square. Both methods provide the same solutions, approximately 1.435 and -10.435, but they offer different insights into the nature of quadratic equations.
The quadratic formula is a direct and efficient method that can be applied to any quadratic equation. It's a versatile tool that guarantees a solution, making it a reliable choice for solving a wide range of problems. The formula itself is a testament to the power of algebraic abstraction, encapsulating the solution to all quadratic equations in a single, elegant expression.
Completing the square, on the other hand, is a more involved method that provides a deeper understanding of the structure of quadratic equations. It reveals the relationship between the equation and its graph, a parabola, and it forms the basis for deriving the quadratic formula. Completing the square is particularly useful for transforming quadratic equations into vertex form, which provides valuable information about the parabola's vertex and axis of symmetry.
By mastering both the quadratic formula and completing the square, you gain a comprehensive toolkit for solving quadratic equations. You also develop a deeper appreciation for the interconnectedness of algebraic concepts. Each method offers a unique perspective on the nature of quadratic equations, and together, they provide a solid foundation for further studies in mathematics.
The quadratic equation x² + 9x - 15 = 0 serves as a good example for illustrating these methods. It has real solutions, but they are not rational, meaning they cannot be expressed as simple fractions. This highlights the importance of having methods like the quadratic formula and completing the square, which can handle equations with irrational solutions. Furthermore, the discriminant (b² - 4ac) of this equation is positive, indicating that there are two distinct real solutions, which we found using both methods.
In conclusion, solving quadratic equations is a fundamental skill in algebra, and the quadratic formula and completing the square are two essential tools for this task. By understanding and mastering these methods, you will be well-equipped to tackle a wide range of mathematical problems and gain a deeper appreciation for the elegance and power of algebraic techniques.
