Quadratic Equation One Real Number Solution

When exploring the fascinating world of mathematics, quadratic equations hold a significant place. These equations, characterized by their highest power of 2, often present a variety of solutions. However, a particularly intriguing scenario arises when a quadratic equation has only one real number solution. In this article, we will delve deep into understanding the conditions under which a quadratic equation possesses a single real root and how to identify such equations.

Understanding Quadratic Equations

Before we dive into the specifics of single-solution quadratic equations, let's first establish a solid understanding of what quadratic equations are and their general form. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually denoted as x) is 2. The general form of a quadratic equation is:

$ax^2 + bx + c = 0$

where a, b, and c are constants, and a is not equal to 0. These constants play a crucial role in determining the nature and number of solutions (also called roots) of the equation. The solutions to a quadratic equation are the values of x that satisfy the equation, making the expression equal to zero. Graphically, these solutions represent the points where the parabola defined by the quadratic equation intersects the x-axis.

The Discriminant: Unveiling the Nature of Roots

The key to understanding the number of real solutions a quadratic equation has lies in a critical component called the discriminant. The discriminant, often denoted by the Greek letter delta (Δ), is a mathematical expression derived from the coefficients of the quadratic equation. It provides valuable information about the nature of the roots without actually solving the equation. The discriminant is calculated as follows:

Δ = $b^2 - 4ac$

The value of the discriminant dictates the number and type of solutions:

• Δ > 0: The equation has two distinct real solutions.
• Δ = 0: The equation has exactly one real solution (a repeated root).
• Δ < 0: The equation has no real solutions; the solutions are complex numbers.

The Condition for One Real Solution

Our focus in this article is on quadratic equations with one real solution. As we learned from the discriminant, this condition is met when Δ = 0. This means that for a quadratic equation to have a single real root, the following must be true:

$b^2 - 4ac = 0$

This equation forms the basis for identifying quadratic equations with exactly one real solution. To determine whether a given quadratic equation has a single real root, we simply need to calculate its discriminant and check if it equals zero.

Identifying Equations with One Real Solution: Practical Examples

Now, let's put our knowledge into practice and examine some example quadratic equations. We'll calculate the discriminant for each equation and determine whether it has one real solution.

Example 1: $0 = 2x^2 - 4x + 1$

In this equation, a = 2, b = -4, and c = 1. Let's calculate the discriminant:

Δ = $(-4)^2 - 4(2)(1) = 16 - 8 = 8$

Since Δ > 0, this equation has two distinct real solutions.

Example 2: $0 = 2x^2 - 5x + 3$

Here, a = 2, b = -5, and c = 3. Calculating the discriminant:

Δ = $(-5)^2 - 4(2)(3) = 25 - 24 = 1$

Since Δ > 0, this equation also has two distinct real solutions.

Example 3: $0 = -2x^2 - 4x - 2$

In this case, a = -2, b = -4, and c = -2. Let's find the discriminant:

Δ = $(-4)^2 - 4(-2)(-2) = 16 - 16 = 0$

Since Δ = 0, this equation has exactly one real solution.

Example 4: $0 = -2x^2 - 3x - 1$

For this equation, a = -2, b = -3, and c = -1. Calculating the discriminant:

Δ = $(-3)^2 - 4(-2)(-1) = 9 - 8 = 1$

Since Δ > 0, this equation has two distinct real solutions.

Conclusion: Mastering the Discriminant for Single-Solution Equations

In conclusion, a quadratic equation has one real number solution if and only if its discriminant (Δ) is equal to zero. The discriminant, calculated as $b^2 - 4ac$, provides a powerful tool for quickly determining the nature of the roots of a quadratic equation without the need to solve the equation itself. By understanding and applying the concept of the discriminant, we can efficiently identify quadratic equations with a single real solution.

Throughout this article, we have explored the fundamental principles of quadratic equations, the significance of the discriminant, and the specific condition required for a quadratic equation to possess one real root. By working through practical examples, we have demonstrated how to apply the discriminant to identify such equations. This knowledge equips us with a valuable tool for solving mathematical problems and gaining a deeper understanding of the behavior of quadratic equations.

Remember, mastering the discriminant is key to unlocking the secrets of quadratic equations and their solutions. With this knowledge, you can confidently tackle a wide range of mathematical challenges involving quadratic equations.

While we have covered the core concepts of quadratic equations with a single real solution, there are many avenues for further exploration. Here are some topics you might find interesting:

1. The Quadratic Formula: While the discriminant tells us the nature of the roots, the quadratic formula allows us to find the actual values of the solutions. Explore how the quadratic formula is derived and how it relates to the discriminant.
2. Graphing Quadratic Equations: The graph of a quadratic equation is a parabola. Investigate how the discriminant affects the shape and position of the parabola, particularly in the case of equations with one real solution.
3. Applications of Quadratic Equations: Quadratic equations have numerous applications in real-world scenarios, such as physics, engineering, and economics. Research how quadratic equations are used to model and solve problems in these fields.
4. Complex Solutions: When the discriminant is negative, the quadratic equation has complex solutions. Dive into the world of complex numbers and learn how to find the complex roots of quadratic equations.
5. Systems of Equations: Explore how quadratic equations can be combined with other equations to form systems of equations. Learn how to solve these systems and interpret their solutions.

By delving deeper into these topics, you can expand your understanding of quadratic equations and their significance in mathematics and beyond.

To solidify your understanding of quadratic equations with a single real solution, it's essential to practice applying the concepts we've discussed. Here are some exercises to challenge yourself:

1. Identify Equations with One Real Solution: Given a set of quadratic equations, determine which ones have exactly one real solution by calculating their discriminants.
2. Construct Equations with One Real Solution: Create your own quadratic equations that have one real solution. Start by choosing values for a and b, then calculate the value of c that makes the discriminant equal to zero.
3. Solve Equations with One Real Solution: For the equations you identified or constructed in the previous exercises, find the single real solution using the quadratic formula or by factoring.
4. Graph Equations with One Real Solution: Graph the quadratic equations you've been working with and observe how the parabola touches the x-axis at only one point.
5. Real-World Applications: Find examples of real-world problems that can be modeled using quadratic equations with one real solution. Solve these problems and interpret the results.

By working through these exercises, you will develop a deeper understanding of quadratic equations with a single real solution and enhance your problem-solving skills.

The journey of understanding quadratic equations and their solutions is a testament to the beauty of mathematical discovery. By exploring the concepts, practicing the techniques, and delving deeper into related topics, you can unlock the power of mathematics and appreciate its elegance and applicability. So, continue your exploration, ask questions, and embrace the challenge of mathematical problem-solving. The world of quadratic equations and beyond awaits your curiosity and ingenuity!