Solving Quadratic Equations With Arithmetic Progression Solutions

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Introduction

In this comprehensive exploration, we delve into the fascinating realm of quadratic equations and their solutions, particularly when those solutions exhibit a unique relationship – they form an arithmetic progression. Our primary focus lies on a specific quadratic equation: x² - (2n + 3)x + 90 = 0. We embark on a journey to unravel the conditions under which the solutions to this equation constitute an arithmetic progression with a common difference of 'n'. Furthermore, we aim to determine the maximum possible difference between these solutions. This exploration will not only enhance our understanding of quadratic equations but also shed light on the interplay between algebra and arithmetic sequences.

Understanding the Fundamentals of Quadratic Equations

Before diving into the specifics of our problem, let's solidify our understanding of quadratic equations. A quadratic equation is a polynomial equation of the second degree, generally represented in the form ax² + bx + c = 0, where a, b, and c are constants, and 'a' is not equal to zero. The solutions to a quadratic equation, also known as its roots, are the values of 'x' that satisfy the equation. These roots can be real or complex numbers.

There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. The quadratic formula is a versatile tool that provides the solutions for any quadratic equation, regardless of its factorability. It is given by:

x = [-b ± √(b² - 4ac)] / 2a

In the context of our problem, the quadratic equation is x² - (2n + 3)x + 90 = 0. By comparing this equation with the general form, we can identify the coefficients: a = 1, b = -(2n + 3), and c = 90. These coefficients will play a crucial role in determining the solutions and their relationship.

The Significance of the Discriminant

Within the quadratic formula, the expression b² - 4ac holds special significance. It is known as the discriminant and provides valuable information about the nature of the roots. The discriminant can be used to classify the roots as follows:

  • If b² - 4ac > 0, the equation has two distinct real roots.
  • If b² - 4ac = 0, the equation has one real root (a repeated root).
  • If b² - 4ac < 0, the equation has two complex roots.

In our quest to find solutions that form an arithmetic progression, we need to ensure that the roots are real. Therefore, the discriminant of our equation must be greater than or equal to zero. This condition will help us establish a constraint on the possible values of 'n'.

Arithmetic Progressions: A Quick Review

An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted by 'd'. The general form of an arithmetic progression is:

a, a + d, a + 2d, a + 3d, ...

where 'a' is the first term. The nth term of an arithmetic progression can be expressed as:

an = a + (n - 1)d

In our problem, we are given that the solutions to the quadratic equation form an arithmetic progression with a common difference of 'n'. This piece of information is crucial in connecting the solutions of the equation with the properties of arithmetic progressions. It implies that if one solution is 'x1', the other solution must be 'x1 + n'. This relationship will be instrumental in setting up equations and solving for 'n'.

Connecting Quadratic Equations and Arithmetic Progressions

Now, let's bridge the gap between quadratic equations and arithmetic progressions. We have the quadratic equation x² - (2n + 3)x + 90 = 0, and we know that its solutions, say x1 and x2, form an arithmetic progression with a common difference of 'n'. This means x2 = x1 + n.

From the theory of quadratic equations, we know the following relationships between the roots and the coefficients:

  • Sum of the roots: x1 + x2 = -b/a
  • Product of the roots: x1 * x2 = c/a

In our case, a = 1, b = -(2n + 3), and c = 90. Therefore,

  • x1 + x2 = 2n + 3
  • x1 * x2 = 90

We now have three equations:

  1. x2 = x1 + n
  2. x1 + x2 = 2n + 3
  3. x1 * x2 = 90

These equations form a system that we can solve to find the values of x1, x2, and n. The solution to this system will provide us with the specific arithmetic progressions that satisfy the given conditions.

Solving the System of Equations

To solve the system of equations, we can use substitution. Substitute equation (1) into equation (2):

x1 + (x1 + n) = 2n + 3

Simplify the equation:

2x1 + n = 2n + 3

Solve for x1:

2x1 = n + 3

x1 = (n + 3) / 2

Now, substitute x1 = (n + 3) / 2 and x2 = x1 + n into equation (3):

[(n + 3) / 2] * [(n + 3) / 2 + n] = 90

Simplify and solve for n:

[(n + 3) / 2] * [(3n + 3) / 2] = 90

(n + 3)(3n + 3) = 360

3n² + 12n + 9 = 360

3n² + 12n - 351 = 0

n² + 4n - 117 = 0

This is a quadratic equation in 'n'. We can solve it by factoring or using the quadratic formula. Factoring yields:

(n - 9)(n + 13) = 0

Therefore, the possible values for 'n' are n = 9 and n = -13. However, since the common difference in an arithmetic progression is usually considered positive, we focus on n = 9.

Determining the Solutions and the Maximum Difference

Now that we have found n = 9, we can find the solutions x1 and x2:

x1 = (n + 3) / 2 = (9 + 3) / 2 = 6

x2 = x1 + n = 6 + 9 = 15

Thus, the solutions to the quadratic equation are 6 and 15, which form an arithmetic progression with a common difference of 9. The difference between the solutions is:

x2 - x1 = 15 - 6 = 9

Since we have only one valid value for 'n' (n = 9), the maximum value of the difference between the solutions is 9.

Conclusion

In this exploration, we successfully navigated the intersection of quadratic equations and arithmetic progressions. We determined that the quadratic equation x² - (2n + 3)x + 90 = 0 has solutions that form an arithmetic progression with a common difference of 'n' when n = 9. The solutions are 6 and 15, and the maximum difference between them is 9. This problem highlights the beauty of mathematics in connecting seemingly disparate concepts and provides a valuable exercise in problem-solving techniques.

This journey through quadratic equations and arithmetic progressions underscores the importance of understanding fundamental concepts and their interrelationships. By applying the quadratic formula, analyzing the discriminant, and leveraging the properties of arithmetic progressions, we were able to solve a challenging problem and gain a deeper appreciation for the elegance of mathematical reasoning.