Solving The Quadratic Equation R^2 - 7r - 8 = 0

Introduction

In the realm of mathematics, solving quadratic equations is a fundamental skill. Quadratic equations, characterized by the general form ax² + bx + c = 0, where a, b, and c are constants, appear in various fields, from physics and engineering to economics and computer science. This article delves into the process of finding the possible value or values of r in the specific quadratic equation r² - 7r - 8 = 0. We will explore different methods to solve this equation, including factoring, using the quadratic formula, and completing the square, ultimately determining the correct solution from the given options. Understanding how to solve quadratic equations is crucial for anyone pursuing studies or careers involving mathematical applications.

Understanding Quadratic Equations

Before diving into the solution, let's establish a solid understanding of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are coefficients, and x is the variable. The coefficients can be any real numbers, with the crucial condition that a cannot be zero; otherwise, the equation would reduce to a linear equation.

The solutions to a quadratic equation are also known as its roots or zeros. These roots represent the values of the variable that satisfy the equation, making the expression equal to zero. A quadratic equation can have two distinct real roots, one repeated real root, or two complex roots. The nature and number of roots are determined by the discriminant, which we will discuss later in the context of the quadratic formula. Understanding the structure and properties of quadratic equations is essential for selecting the most appropriate method for solving them and interpreting the results.

Methods to Solve Quadratic Equations

Several methods can be employed to solve quadratic equations, each with its own advantages and suitability depending on the specific equation. The most common methods include:

• Factoring: This method involves expressing the quadratic expression as a product of two linear factors. Factoring is often the quickest method when the equation can be easily factored.
• Quadratic Formula: The quadratic formula provides a general solution for any quadratic equation, regardless of its factorability. It is particularly useful when factoring is difficult or impossible.
• Completing the Square: This method involves transforming the quadratic equation into a perfect square trinomial, which can then be easily solved by taking the square root. Completing the square is a foundational technique that also underlies the derivation of the quadratic formula.

Each method offers a unique approach to finding the roots of a quadratic equation, and proficiency in all three methods provides a comprehensive toolkit for solving a wide range of problems. In the following sections, we will apply these methods to the given equation, r² - 7r - 8 = 0, to determine the correct solution.

Solving r² - 7r - 8 = 0 by Factoring

Factoring is a straightforward method for solving quadratic equations when the expression can be easily decomposed into two linear factors. This method hinges on finding two numbers that multiply to the constant term (c) and add up to the coefficient of the linear term (b). In the given equation, r² - 7r - 8 = 0, we need to find two numbers that multiply to -8 and add up to -7.

By considering the factors of -8, we can identify the numbers -8 and 1 as the pair that satisfies these conditions: (-8) * (1) = -8 and (-8) + 1 = -7. Therefore, we can rewrite the quadratic equation by splitting the middle term:

• r² - 8r + 1r - 8 = 0

Now, we can factor by grouping:

• r(r - 8) + 1(r - 8) = 0

Notice that (r - 8) is a common factor. Factoring it out, we get:

• (r - 8)(r + 1) = 0

For the product of two factors to be zero, at least one of them must be zero. This leads to two possible solutions:

• r - 8 = 0 => r = 8
• r + 1 = 0 => r = -1

Thus, the solutions to the quadratic equation r² - 7r - 8 = 0 by factoring are r = 8 and r = -1. This method highlights the elegance and efficiency of factoring when applicable.

Solving r² - 7r - 8 = 0 Using the Quadratic Formula

When factoring is not readily apparent or proves difficult, the quadratic formula offers a reliable and universally applicable method for solving quadratic equations. The quadratic formula provides the solutions for any equation in the form ax² + bx + c = 0:

• r = (-b ± √(b² - 4ac)) / 2a

In our case, the equation r² - 7r - 8 = 0 has coefficients a = 1, b = -7, and c = -8. Substituting these values into the quadratic formula, we get:

• r = (-(-7) ± √((-7)² - 4 * 1 * (-8))) / (2 * 1)
• r = (7 ± √(49 + 32)) / 2
• r = (7 ± √81) / 2
• r = (7 ± 9) / 2

This gives us two possible solutions:

• r = (7 + 9) / 2 = 16 / 2 = 8
• r = (7 - 9) / 2 = -2 / 2 = -1

Therefore, using the quadratic formula, we find the solutions to the equation r² - 7r - 8 = 0 to be r = 8 and r = -1, consistent with the solutions obtained by factoring. This method demonstrates the power and generality of the quadratic formula in solving any quadratic equation.

Identifying the Correct Solution

Having solved the quadratic equation r² - 7r - 8 = 0 using both factoring and the quadratic formula, we have consistently arrived at the solutions r = 8 and r = -1. Now, let's compare these solutions with the given options:

A) r = (17 + √277) / 6, r = (17 - √277) / 6 B) r = -10, r = 3 C) r = 8, r = -1 D) r = 2 / 3, r = 5

By direct comparison, it is evident that option C, r = 8, r = -1, matches the solutions we obtained through our calculations. Therefore, option C is the correct answer. This step underscores the importance of verifying the solutions obtained against the provided options to ensure accuracy.

Conclusion

In summary, we successfully found the possible values of r in the quadratic equation r² - 7r - 8 = 0 by employing two distinct methods: factoring and the quadratic formula. Both methods yielded the same solutions, r = 8 and r = -1. By comparing these results with the given options, we identified option C as the correct answer. This exercise highlights the significance of understanding and applying different techniques for solving quadratic equations, reinforcing the foundational role of these skills in mathematics and its applications. The ability to solve quadratic equations is a valuable asset in various fields, enabling the analysis and modeling of real-world phenomena. Proficiency in these methods empowers individuals to approach mathematical challenges with confidence and precision.