solution with explaination....................................................

To provide a comprehensive solution and explanation, let’s break down the problem step by step. It’s essential to understand the context and the specific details of the question to arrive at a well-reasoned answer. For this example, let’s assume we’re discussing a mathematical problem involving quadratic equations.

Understanding Quadratic Equations

A quadratic equation is typically in the form of ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The solutions to this equation can be found using various methods, including factoring, completing the square, or applying the quadratic formula.

The Quadratic Formula

The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

This formula provides the values of x that satisfy the equation. The term under the square root, b² - 4ac, is known as the discriminant. It tells us about the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root (a repeated root).
• If it is negative, the roots are complex (not real).

Example Problem

Let’s solve the quadratic equation 2x² - 4x - 6 = 0 using the quadratic formula.

Step 1: Identify the coefficients

Here, we have:

• a = 2
• b = -4
• c = -6

Step 2: Calculate the discriminant

Now, we calculate the discriminant:

b² - 4ac = (-4)² - 4(2)(-6) = 16 + 48 = 64

Since the discriminant is positive, we expect two distinct real roots.

Step 3: Apply the quadratic formula

Now, substituting the values into the quadratic formula:

x = (4 ± √64) / (2 * 2)

This simplifies to:

x = (4 ± 8) / 4

Step 4: Find the roots

Calculating the two possible values:

• x₁ = (4 + 8) / 4 = 12 / 4 = 3
• x₂ = (4 - 8) / 4 = -4 / 4 = -1

Final Thoughts

The solutions to the equation 2x² - 4x - 6 = 0 are x = 3 and x = -1. This method not only provides the answers but also illustrates the importance of understanding each step in the process. By breaking down the problem, we can tackle even the most complex equations with confidence.