Using The Discriminant To Find Solutions Of Quadratic Equation 25x^2 + 10x + 1 = 0

In the realm of mathematics, quadratic equations hold a significant place, appearing in various fields from physics to engineering. Understanding the nature of their solutions is crucial, and the discriminant serves as a powerful tool in this endeavor. This article delves into the concept of the discriminant and how it helps determine the number and type of solutions for a given quadratic equation. We will specifically explore the equation 25x² + 10x + 1 = 0 and apply the discriminant to unravel its solution characteristics.

What is a Quadratic Equation?

Before diving into the discriminant, it's essential to understand what constitutes a quadratic equation. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is:

ax² + bx + c = 0

where 'a', 'b', and 'c' are constants, and 'x' is the variable. The coefficient 'a' cannot be zero, otherwise, the equation would reduce to a linear equation.

Quadratic equations can have up to two solutions, also known as roots or zeros. These solutions represent the values of 'x' that satisfy the equation, making the left-hand side equal to zero. These solutions can be real or complex numbers.

The Quadratic Formula: Unveiling the Solutions

The quadratic formula is a fundamental tool for finding the solutions of a quadratic equation. It provides a direct method to calculate the roots based on the coefficients 'a', 'b', and 'c'. The quadratic formula is expressed as:

x = (-b ± √(b² - 4ac)) / 2a

This formula reveals two potential solutions, distinguished by the plus-minus (±) sign. The expression inside the square root, b² - 4ac, plays a pivotal role in determining the nature of the solutions. This expression is known as the discriminant.

The Discriminant: A Key to Understanding Solutions

The discriminant, denoted as Δ (Delta), is the expression b² - 4ac within the quadratic formula. It acts as a critical indicator, revealing the number and type of solutions a quadratic equation possesses without actually solving the equation. The discriminant's value dictates whether the solutions are real and distinct, real and equal, or complex.

Interpreting the Discriminant's Value

1. Δ > 0 (Positive Discriminant): When the discriminant is positive, the quadratic equation has two distinct real solutions. This means the parabola represented by the equation intersects the x-axis at two different points. The square root of a positive number yields two real values, leading to two different solutions when plugged into the quadratic formula. Understanding these distinct real roots is fundamental in various applications.
2. Δ = 0 (Zero Discriminant): If the discriminant is zero, the quadratic equation has exactly one real solution (or two equal real solutions). In this case, the parabola touches the x-axis at only one point. The square root of zero is zero, causing the plus-minus part of the quadratic formula to vanish, resulting in a single solution. This single real root scenario is crucial in optimization problems.
3. Δ < 0 (Negative Discriminant): When the discriminant is negative, the quadratic equation has two complex solutions. These solutions involve imaginary numbers (numbers that include the square root of -1, denoted as 'i'). The parabola does not intersect the x-axis in this case. The square root of a negative number results in an imaginary number, leading to complex solutions. Grasping complex roots is essential in advanced mathematical contexts.

Applying the Discriminant to 25x² + 10x + 1 = 0

Now, let's apply the concept of the discriminant to the given quadratic equation: 25x² + 10x + 1 = 0. First, we identify the coefficients:

• a = 25
• b = 10
• c = 1

Next, we calculate the discriminant using the formula:

Δ = b² - 4ac

Substituting the values, we get:

Δ = (10)² - 4 * 25 * 1

Δ = 100 - 100

Δ = 0

The discriminant is 0. This indicates that the quadratic equation 25x² + 10x + 1 = 0 has exactly one real solution (or two equal real solutions).

Finding the Solution

Since the discriminant is zero, we know there is one real solution. We can find this solution using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Substituting the values, we get:

x = (-10 ± √0) / (2 * 25)

x = -10 / 50

x = -1/5

Therefore, the quadratic equation 25x² + 10x + 1 = 0 has one real solution, which is x = -1/5. This implies that the parabola represented by the equation touches the x-axis at the single point x = -1/5. Calculating this real solution confirms the discriminant's prediction.

Conclusion

The discriminant is a powerful tool for quickly determining the nature of solutions to a quadratic equation. By calculating b² - 4ac, we can predict whether the equation has two distinct real solutions, one real solution (or two equal real solutions), or two complex solutions. Applying this concept to the equation 25x² + 10x + 1 = 0, we found that the discriminant is 0, indicating one real solution. Further calculations using the quadratic formula confirmed this, revealing the solution to be x = -1/5. Understanding the discriminant enhances our ability to analyze and solve quadratic equations efficiently, paving the way for tackling more complex mathematical problems.

By understanding the role of the discriminant, mathematicians, engineers, and scientists can efficiently analyze quadratic equations, determine the nature of their solutions, and apply this knowledge to various real-world problems. This fundamental concept is vital for anyone seeking a deeper understanding of mathematical principles.