Nature Of Solutions Analyzing Quadratic Equation Y=x²-11x+7

When delving into the realm of quadratic equations, understanding the nature of their solutions is paramount. The equation presented, y=x²-11x+7, serves as an excellent example to explore the concepts of real and complex roots. Determining whether a quadratic equation has one or two solutions, and whether these solutions are real or complex, hinges on a critical component: the discriminant. This article will meticulously analyze the given quadratic equation, guiding you through the process of calculating the discriminant and interpreting its value to ascertain the nature of the solutions. We'll dissect each answer choice, providing a comprehensive understanding of why a particular option is correct or incorrect. By the end of this discussion, you'll not only be able to confidently answer this specific question but also possess a robust foundation for tackling similar problems involving quadratic equations and their solutions.

At the heart of understanding the nature of solutions in a quadratic equation lies the discriminant. This seemingly simple value, derived from the coefficients of the quadratic equation, acts as a powerful indicator of the roots' characteristics. To fully grasp its significance, let's first revisit the standard form of a quadratic equation: ax² + bx + c = 0. The discriminant, often denoted by the Greek letter delta (Δ), is calculated using the formula: Δ = b² - 4ac. This formula is not just a mathematical curiosity; it's the key to unlocking the secrets of the solutions.

The discriminant's value dictates the number and type of solutions the quadratic equation possesses. Here's a breakdown of how the discriminant influences the solutions:

• Δ > 0 (Positive Discriminant): This indicates that the quadratic equation has two distinct real solutions. These solutions represent the points where the parabola intersects the x-axis on a graph.
• Δ = 0 (Zero Discriminant): In this case, the equation has one real solution (or, more accurately, two identical real solutions). This signifies that the parabola touches the x-axis at a single point.
• Δ < 0 (Negative Discriminant): This signifies that the equation has two complex solutions. Complex solutions involve the imaginary unit 'i' (where i² = -1) and arise when the parabola does not intersect the x-axis.

Understanding the discriminant is not just about memorizing a formula; it's about grasping the fundamental relationship between the coefficients of a quadratic equation and the nature of its roots. This knowledge empowers us to analyze any quadratic equation and predict the characteristics of its solutions without explicitly solving for them.

Now, let's put our knowledge of the discriminant to the test by analyzing the given quadratic equation: y = x² - 11x + 7. To determine the nature of its solutions, we need to identify the coefficients a, b, and c, and then calculate the discriminant. Comparing the equation to the standard form ax² + bx + c = 0, we can readily identify the coefficients:

• a = 1 (the coefficient of x²)
• b = -11 (the coefficient of x)
• c = 7 (the constant term)

With these values in hand, we can now calculate the discriminant using the formula Δ = b² - 4ac. Substituting the values, we get:

Δ = (-11)² - 4 * 1 * 7 Δ = 121 - 28 Δ = 93

The discriminant, Δ, is calculated to be 93. This is a positive value. Remember our earlier discussion about the implications of the discriminant's value? A positive discriminant indicates that the quadratic equation has two distinct real solutions. This means the parabola represented by the equation intersects the x-axis at two different points.

By systematically applying the concept of the discriminant, we've successfully determined the nature of the solutions for the given quadratic equation. This step-by-step analysis highlights the power of the discriminant as a tool for understanding the roots of quadratic equations.

Having calculated the discriminant and determined that the quadratic equation y = x² - 11x + 7 has two real solutions, we can now evaluate the answer choices provided and identify the correct one. Let's examine each option in light of our findings:

• A. There is one complex solution. This statement is incorrect. Our calculated discriminant (Δ = 93) is positive, indicating real solutions, not complex solutions. Complex solutions arise when the discriminant is negative.
• B. There are two real solutions. This statement aligns perfectly with our calculated discriminant. A positive discriminant confirms the existence of two distinct real solutions. Therefore, this is the correct answer.
• C. There is one real solution. This statement is incorrect. A discriminant of 93, being positive, signifies two real solutions, not one.
• D. There are two complex solutions. This statement is also incorrect. As we established, a positive discriminant rules out the possibility of complex solutions.

Therefore, the correct answer is B. There are two real solutions. This process of elimination, guided by our understanding of the discriminant, reinforces the importance of this concept in analyzing quadratic equations.

To further solidify our understanding, let's delve deeper into the distinction between real and complex solutions. Real solutions are numbers that can be plotted on a number line; they encompass all rational and irrational numbers. In the context of quadratic equations, real solutions represent the x-intercepts of the parabola, the points where the graph intersects the x-axis.

Complex solutions, on the other hand, involve the imaginary unit 'i', defined as the square root of -1. They take the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit. Complex solutions arise when the discriminant is negative, indicating that the parabola does not intersect the x-axis. These solutions exist in a different mathematical space, often visualized on a complex plane rather than a simple number line.

The presence of complex solutions signifies that the quadratic equation has no real roots. This doesn't mean the equation is unsolvable; it simply means the solutions lie outside the realm of real numbers. Understanding the difference between real and complex solutions is crucial for accurately interpreting the results of quadratic equations and their applications in various fields, such as physics and engineering.

Quadratic equations are not just abstract mathematical concepts; they have profound applications in the real world. Their ability to model parabolic trajectories makes them invaluable in fields like physics, where they are used to describe the motion of projectiles, such as a ball thrown through the air or a rocket launched into space. The solutions to these equations can determine the range, maximum height, and time of flight of these projectiles.

In engineering, quadratic equations are essential for designing structures like bridges and arches, ensuring their stability and load-bearing capacity. The parabolic shape, often dictated by quadratic equations, provides optimal structural integrity. Furthermore, quadratic equations find applications in various optimization problems, such as maximizing profit or minimizing costs in business and economics.

The significance of quadratic equations extends beyond specific applications. They serve as a fundamental building block in mathematics, laying the groundwork for more advanced concepts. Their study hones critical thinking and problem-solving skills, essential for success in various academic and professional pursuits. By mastering quadratic equations, we not only gain a powerful mathematical tool but also cultivate valuable analytical abilities.

In this comprehensive exploration, we dissected the quadratic equation y = x² - 11x + 7, unraveling the nature of its solutions through the lens of the discriminant. We established that the discriminant, calculated as 93, is positive, indicating the presence of two distinct real solutions. This led us to confidently select answer choice B as the correct solution. We further expanded our understanding by differentiating between real and complex solutions and highlighting the wide-ranging applications of quadratic equations in real-world scenarios.

Mastering the concepts surrounding quadratic equations, particularly the discriminant, empowers us to analyze and interpret mathematical relationships effectively. This knowledge not only helps in solving specific problems but also fosters a deeper appreciation for the power and elegance of mathematics in modeling the world around us. Remember, the discriminant is your key to unlocking the secrets of quadratic equations – use it wisely, and you'll be well-equipped to tackle any quadratic challenge that comes your way.