Determining The Nature Of Solutions For The Quadratic Equation Y Equals X Squared Minus 11x Plus 7

When delving into the realm of mathematics, quadratic equations hold a significant position. These equations, characterized by their highest power being two, appear in various applications, ranging from physics to engineering. One of the fundamental aspects of a quadratic equation is understanding the nature of its solutions. In this article, we will explore the quadratic equation $y = x^2 - 11x + 7$ and determine the true statement regarding its solutions. Specifically, we aim to identify whether the equation possesses one real solution, two real solutions, one complex solution, or two complex solutions. To achieve this, we will employ a powerful tool known as the discriminant.

The discriminant is a crucial component derived from the quadratic formula, which provides valuable insights into the nature of the roots of a quadratic equation. The quadratic formula, given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

serves as a universal method for solving quadratic equations of the form $ax^2 + bx + c = 0$. The expression within the square root, $b^2 - 4ac$, is the discriminant. It acts as a determinant, revealing the characteristics of the equation's solutions. The discriminant, often denoted as Δ, dictates whether the quadratic equation has real or complex solutions and how many of each type exist. Understanding the discriminant allows us to predict the nature of the roots without explicitly solving the equation, saving time and effort. By calculating the discriminant for the given equation, we can accurately classify its solutions and gain a deeper understanding of its mathematical behavior.

To begin our analysis, let's identify the coefficients of the quadratic equation $y = x^2 - 11x + 7$. In this equation, we have:

• $a = 1$ (the coefficient of $x^2$)
• $b = -11$ (the coefficient of $x$)
• $c = 7$ (the constant term)

With these coefficients in hand, we can now calculate the discriminant (Δ) using the formula:

$Δ = b^2 - 4ac$

Substituting the values, we get:

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

The discriminant, Δ, is 93. This positive value provides us with crucial information about the nature of the solutions to the quadratic equation. A positive discriminant signifies that the quadratic equation has two distinct real solutions. This is because the square root of a positive number is a real number, and the $\pm$ sign in the quadratic formula leads to two different values for $x$. Therefore, based on the positive discriminant we calculated, we can confidently conclude that the quadratic equation $y = x^2 - 11x + 7$ has two real solutions. This understanding is fundamental in various mathematical applications, such as graphing quadratic functions and solving related problems.

Delving Deeper: The Significance of the Discriminant

To fully grasp the implications of our findings, let's delve deeper into the significance of the discriminant. As we've established, the discriminant (Δ) is the expression $b^2 - 4ac$ found under the square root in the quadratic formula. Its value dictates the nature and number of solutions to the quadratic equation $ax^2 + bx + c = 0$. The discriminant serves as a powerful indicator, allowing us to classify quadratic equations into distinct categories based on their solution characteristics. Understanding the relationship between the discriminant and the solutions is essential for solving quadratic equations efficiently and accurately. Furthermore, this knowledge extends to various mathematical and scientific fields where quadratic equations are frequently encountered.

When the discriminant is positive (Δ > 0), as in our example with $y = x^2 - 11x + 7$, the quadratic equation has two distinct real solutions. This is because the square root of a positive number is a real number, and the $\pm$ sign in the quadratic formula generates two different real values for $x$. Graphically, this means the parabola represented by the quadratic equation intersects the x-axis at two distinct points. These intersection points correspond to the two real solutions of the equation. The further apart these points are, the greater the value of the discriminant, indicating a wider spread of the roots along the x-axis. Therefore, a positive discriminant is a clear indicator of two real solutions, enabling us to visualize and interpret the behavior of the quadratic equation.

Conversely, when the discriminant is zero (Δ = 0), the quadratic equation has exactly one real solution. In this scenario, the square root of the discriminant is zero, and the $\pm$ sign in the quadratic formula does not produce two different values. The result is a single real solution, often referred to as a repeated root. Graphically, this corresponds to the parabola touching the x-axis at only one point, its vertex. This point represents the single real solution of the equation. A zero discriminant signifies a critical point where the parabola transitions from opening upwards to downwards, or vice versa. This special case is essential in various applications, such as optimization problems where finding the minimum or maximum value of a quadratic function is crucial.

Lastly, when the discriminant is negative (Δ < 0), the quadratic equation has two complex solutions. This occurs because the square root of a negative number is not a real number; it is an imaginary number. Complex solutions involve the imaginary unit $i$, where $i^2 = -1$. The quadratic formula yields two complex solutions that are complex conjugates of each other, meaning they have the form $a + bi$ and $a - bi$, where $a$ and $b$ are real numbers. Graphically, this implies that the parabola does not intersect the x-axis at any point. The complex solutions represent points in the complex plane, extending the concept of solutions beyond the real number line. Complex solutions are vital in advanced mathematics, physics, and engineering, particularly in areas involving oscillations, waves, and electrical circuits.

Connecting the Dots: From Discriminant to Solutions

Let's solidify our understanding by connecting the dots between the discriminant and the solutions of a quadratic equation. We've established that the discriminant (Δ) is a critical value that determines the nature and number of solutions. To reiterate, a positive discriminant (Δ > 0) indicates two distinct real solutions, a zero discriminant (Δ = 0) signifies one real solution, and a negative discriminant (Δ < 0) implies two complex solutions. These relationships are fundamental in solving quadratic equations and interpreting their results. By calculating the discriminant first, we can predict the type of solutions we will encounter, which can streamline the solving process and prevent unnecessary calculations.

The connection between the discriminant and the solutions is further highlighted by the quadratic formula itself:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The discriminant, $b^2 - 4ac$, is nestled under the square root. This position is crucial because the square root operation dictates the nature of the result. If the discriminant is positive, the square root yields a real number, leading to two real solutions due to the $\pm$ sign. If the discriminant is zero, the square root is zero, resulting in one real solution. If the discriminant is negative, the square root produces an imaginary number, giving rise to two complex solutions. Thus, the discriminant acts as the gatekeeper, determining whether the solutions reside in the realm of real numbers or venture into the complex plane.

Understanding this connection allows us to tackle quadratic equations with greater confidence. For instance, if we are presented with a quadratic equation and asked to find its solutions, our first step should be to calculate the discriminant. This initial step provides valuable information about the expected outcome. If the discriminant is positive, we know we're dealing with two real solutions, and we can proceed to find them using the quadratic formula or factoring. If the discriminant is zero, we anticipate one real solution, which can be easily found. If the discriminant is negative, we know we'll encounter complex solutions, requiring us to work with imaginary numbers. This predictive ability not only saves time but also enhances our problem-solving skills.

Moreover, the discriminant provides insights into the graphical representation of quadratic equations. As we've mentioned, a positive discriminant corresponds to the parabola intersecting the x-axis at two points, a zero discriminant means the parabola touches the x-axis at one point, and a negative discriminant indicates that the parabola does not intersect the x-axis. These graphical interpretations provide a visual understanding of the solutions, making the connection between algebra and geometry more apparent. By understanding the discriminant, we can quickly sketch the graph of a quadratic equation, identify its key features, and relate them to the solutions.

Conclusion: The Power of the Discriminant

In conclusion, the statement that is true about the quadratic equation $y = x^2 - 11x + 7$ is B. There are two real solutions. This determination was made by calculating the discriminant, which turned out to be positive. The discriminant is a powerful tool in the analysis of quadratic equations, providing valuable information about the nature and number of solutions. By understanding the relationship between the discriminant and the solutions, we can efficiently solve quadratic equations and gain deeper insights into their mathematical behavior. The discriminant's ability to predict the type of solutions, whether real or complex, makes it an indispensable tool in mathematics, science, and engineering. Its connection to the graphical representation of quadratic equations further enhances its significance, providing a visual understanding of the solutions.

Mastering the concept of the discriminant unlocks a deeper understanding of quadratic equations and their applications. It empowers us to approach these equations with confidence and solve them efficiently. The discriminant serves as a bridge, connecting the algebraic and graphical aspects of quadratic equations, providing a comprehensive view of their solutions. As we continue our mathematical journey, the knowledge of the discriminant will undoubtedly prove invaluable in various problem-solving scenarios.