Finding The Sum Of Roots Of Quadratic Equation X² - 8x + 7 = 0

In the realm of mathematics, quadratic equations hold a significant position, serving as fundamental tools in various fields ranging from physics to engineering. Understanding the properties of these equations, such as the sum and product of their roots, is crucial for problem-solving and analytical purposes. This article delves into the specifics of the quadratic equation x² - 8x + 7 = 0, focusing on determining the sum of its roots. We will explore different methods to arrive at the solution, ensuring a comprehensive understanding of the underlying concepts.

What are Quadratic Equations?

Before we tackle the specific equation, let's establish a firm understanding of quadratic equations in general. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is expressed as:

ax² + bx + c = 0

where 'a', 'b', and 'c' are constants, with 'a' not equal to zero. The solutions to this equation, often denoted as 'x', are called the roots or zeros of the quadratic equation. These roots represent the values of 'x' that satisfy the equation, making the expression equal to zero.

Key Components of a Quadratic Equation

• a: The coefficient of the x² term. This coefficient determines the parabola's concavity (whether it opens upwards or downwards). If 'a' is positive, the parabola opens upwards, and if 'a' is negative, it opens downwards.
• b: The coefficient of the x term. This coefficient influences the axis of symmetry of the parabola.
• c: The constant term. This term represents the y-intercept of the parabola, the point where the parabola intersects the y-axis.

Methods to Find the Roots of a Quadratic Equation

There are several methods to find the roots of a quadratic equation, each with its own advantages and applicability. Let's briefly discuss the most common methods:

1. Factoring

Factoring involves expressing the quadratic equation as a product of two binomials. This method is efficient when the quadratic expression can be easily factored. For example, if the equation is x² - 5x + 6 = 0, it can be factored as (x - 2)(x - 3) = 0. Setting each factor equal to zero gives the roots x = 2 and x = 3.

2. Quadratic Formula

The quadratic formula is a universal method that can be used to find the roots of any quadratic equation, regardless of whether it can be factored easily. The formula is given by:

x = (-b ± √(b² - 4ac)) / 2a

where 'a', 'b', and 'c' are the coefficients of the quadratic equation. The discriminant, b² - 4ac, determines the nature of the roots. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root (a repeated root); and if it is negative, there are two complex roots.

3. Completing the Square

Completing the square involves manipulating the quadratic equation to form a perfect square trinomial. This method is useful for deriving the quadratic formula and for solving equations where factoring is not straightforward. The process involves adding and subtracting a constant term to create a perfect square trinomial on one side of the equation.

Analyzing the Equation x² - 8x + 7 = 0

Now, let's focus on the specific equation in question: x² - 8x + 7 = 0. This equation is a classic example of a quadratic equation, where:

• a = 1
• b = -8
• c = 7

Our primary goal is to determine the sum of the roots of this equation. We will explore two primary methods to achieve this: factoring and using the relationship between the roots and coefficients of a quadratic equation.

Method 1: Factoring the Quadratic Equation

The first approach involves factoring the quadratic equation. We need to find two numbers that multiply to 7 (the constant term) and add up to -8 (the coefficient of the x term). These numbers are -1 and -7. Therefore, we can factor the equation as:

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

Setting each factor equal to zero, we get:

x - 1 = 0 => x = 1 x - 7 = 0 => x = 7

So, the roots of the equation are x = 1 and x = 7.

To find the sum of the roots, we simply add them together:

Sum of roots = 1 + 7 = 8

Method 2: Using the Relationship Between Roots and Coefficients

An alternative method to find the sum of the roots involves using the direct relationship between the roots and coefficients of a quadratic equation. For a quadratic equation in the form ax² + bx + c = 0, the sum of the roots (let's denote them as α and β) is given by:

Sum of roots (α + β) = -b/a

In our equation, x² - 8x + 7 = 0, we have a = 1 and b = -8. Applying the formula, we get:

Sum of roots = -(-8)/1 = 8

This method provides a quick and efficient way to find the sum of the roots without explicitly solving for the roots themselves. It's a powerful tool for problem-solving and can save time in various mathematical contexts.

Significance of the Sum of Roots

The sum of the roots of a quadratic equation is not just a mathematical curiosity; it has significant implications and applications. Understanding this concept helps in:

• Verifying Solutions: Knowing the sum of the roots allows for a quick check of the solutions obtained by factoring or using the quadratic formula. If the sum of the calculated roots does not match the value obtained using the formula -b/a, it indicates a potential error in the solution process.
• Constructing Quadratic Equations: If the roots of a quadratic equation are known, the sum and product of the roots can be used to construct the equation itself. This is particularly useful in reverse-engineering problems where the roots are given, and the equation needs to be determined.
• Analyzing Quadratic Functions: The sum of the roots, along with the product, provides insights into the behavior and characteristics of the quadratic function represented by the equation. These properties are essential in graphing and analyzing quadratic functions.

Conclusion

In summary, the sum of the roots of the quadratic equation x² - 8x + 7 = 0 is 8. We arrived at this solution using two distinct methods: factoring the equation and applying the relationship between the roots and coefficients. Both methods demonstrate the fundamental principles of quadratic equations and their properties. The ability to determine the sum of the roots is a valuable skill in mathematics, with applications in various problem-solving scenarios and analytical contexts. By mastering these concepts, one can gain a deeper understanding of quadratic equations and their role in mathematical analysis. Understanding the sum of roots is crucial not only for academic purposes but also for practical applications in fields such as engineering, physics, and computer science. It forms a cornerstone of algebraic knowledge and enhances the ability to solve complex problems effectively.