Real Number Solutions For Quadratic Function Y = 16x² + 16x + 4

To determine the number of real-number solutions for the quadratic function y = 16x² + 16x + 4, we can use the discriminant. This powerful tool, derived from the quadratic formula, allows us to predict the nature and number of roots (or solutions) of a quadratic equation without actually solving for them. This article will thoroughly explain how to use the discriminant, apply it to the given quadratic function, and interpret the result to find the number of real-number solutions. We will also delve into the background of the discriminant and its relationship to the quadratic formula, providing a comprehensive understanding of this important concept in algebra.

Understanding the Discriminant

When dealing with quadratic equations, the discriminant is a crucial concept to grasp. It serves as a key indicator of the nature and number of solutions a quadratic equation possesses. Before we dive into the specifics of our problem, let's establish a solid understanding of what the discriminant is and why it's so valuable. A quadratic equation is generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and x is the variable we aim to solve for. The discriminant, often denoted by the Greek letter delta (Δ), is calculated using the formula Δ = b² - 4ac. This simple expression holds the key to unlocking the secrets of a quadratic equation's solutions. The discriminant is derived directly from the quadratic formula, which is used to find the solutions (or roots) of any quadratic equation. The quadratic formula is given by: x = (-b ± √(b² - 4ac)) / (2a). Notice that the expression under the square root is precisely the discriminant. This is no coincidence; the discriminant's value dictates the nature of the roots because the square root function behaves differently depending on whether its argument is positive, negative, or zero. To elaborate, the term b² - 4ac within the square root determines whether the solutions are real and distinct, real and equal, or complex (non-real). The beauty of the discriminant lies in its ability to provide this information without the need to fully solve the quadratic equation, saving time and effort. By simply calculating b² - 4ac, we gain immediate insight into the equation's solution landscape. In essence, the discriminant acts as a diagnostic tool, offering a quick and efficient way to classify the roots of a quadratic equation. This understanding is fundamental for solving a wide range of mathematical problems and is a cornerstone of quadratic equation analysis. Understanding the discriminant not only simplifies problem-solving but also enhances our appreciation of the intricate relationships within mathematical formulas.

Calculating the Discriminant for y = 16x² + 16x + 4

Now, let's apply the concept of the discriminant to the given quadratic function, y = 16x² + 16x + 4. To determine the number of real-number solutions, we need to calculate the discriminant (b² - 4ac) for this specific equation. Recall that the general form of a quadratic equation is ax² + bx + c = 0. To use the discriminant, we first need to identify the coefficients a, b, and c in our equation. In the function y = 16x² + 16x + 4, we can see that a is the coefficient of the x² term, b is the coefficient of the x term, and c is the constant term. Therefore, we have: a = 16, b = 16, and c = 4. Now that we have identified the coefficients, we can plug them into the discriminant formula: Δ = b² - 4ac. Substituting the values we found, we get: Δ = (16)² - 4(16)(4). Next, we perform the calculations step by step. First, we square 16: 16² = 256. Then, we multiply 4, 16, and 4: 4 * 16 * 4 = 256. Now, we substitute these values back into the discriminant formula: Δ = 256 - 256. Finally, we subtract to find the value of the discriminant: Δ = 0. The discriminant for the quadratic function y = 16x² + 16x + 4 is 0. This result is crucial because the value of the discriminant tells us a lot about the nature of the solutions to the quadratic equation. In the next section, we will interpret this result and see what it implies about the number of real-number solutions the equation has. By systematically calculating the discriminant, we are able to gain valuable insight into the behavior of the quadratic function without needing to solve the entire equation. This approach is both efficient and insightful, making the discriminant a powerful tool in quadratic equation analysis.

Interpreting the Discriminant Value

Having calculated the discriminant for the quadratic function y = 16x² + 16x + 4, we found that Δ = 0. Now, the crucial step is to interpret this value. The discriminant serves as a powerful indicator of the nature and number of real-number solutions a quadratic equation possesses. There are three key scenarios to consider when interpreting the discriminant: 1. If Δ > 0 (Discriminant is positive): This indicates that the quadratic equation has two distinct real-number solutions. Graphically, this means the parabola represented by the quadratic function intersects the x-axis at two different points. The positive discriminant implies that the square root in the quadratic formula (√(b² - 4ac)) will yield a real, non-zero value, leading to two different solutions when adding and subtracting this value from -b. 2. If Δ = 0 (Discriminant is zero): This indicates that the quadratic equation has exactly one real-number solution (also known as a repeated or double root). In this case, the parabola touches the x-axis at only one point, which is the vertex of the parabola. When the discriminant is zero, the square root in the quadratic formula becomes zero (√0 = 0), and the ± part of the formula vanishes. This results in a single solution because -b / (2a) will be the only root. 3. If Δ < 0 (Discriminant is negative): This indicates that the quadratic equation has no real-number solutions. Instead, it has two complex solutions. Graphically, the parabola does not intersect the x-axis at any point. A negative discriminant means we are taking the square root of a negative number (√(b² - 4ac)), which results in an imaginary number. Therefore, the solutions will involve imaginary numbers, and there will be no real roots. In our specific case, we found that Δ = 0 for the quadratic function y = 16x² + 16x + 4. Based on the above interpretations, this means the quadratic function has exactly one real-number solution. The parabola representing this function touches the x-axis at only one point. Understanding these three scenarios allows us to quickly and accurately determine the nature of the solutions for any quadratic equation by simply calculating and interpreting the discriminant. This powerful tool saves time and provides a deep understanding of the behavior of quadratic functions.

Conclusion: Number of Real-Number Solutions

In conclusion, by using the discriminant, we have successfully determined the number of real-number solutions for the quadratic function y = 16x² + 16x + 4. We began by understanding the significance of the discriminant and its role in determining the nature of the solutions of a quadratic equation. The discriminant, calculated as Δ = b² - 4ac, provides valuable information about whether the quadratic equation has two distinct real solutions, one real solution (a repeated root), or no real solutions (two complex solutions). Applying this knowledge to our specific function, we identified the coefficients a, b, and c as 16, 16, and 4, respectively. We then calculated the discriminant: Δ = (16)² - 4(16)(4) = 256 - 256 = 0. The result, Δ = 0, is of particular significance. As we discussed, a discriminant of zero indicates that the quadratic equation has exactly one real-number solution. This means the parabola represented by the function y = 16x² + 16x + 4 touches the x-axis at only one point, representing a single, repeated root. Therefore, the final answer to the question, "How many real-number solutions does this quadratic function have?" is that it has one real-number solution. This process demonstrates the power and efficiency of using the discriminant to analyze quadratic equations. Instead of having to solve the entire equation using the quadratic formula, we were able to quickly determine the number of real solutions by simply calculating the discriminant. This is a fundamental concept in algebra and is essential for understanding the behavior of quadratic functions. By mastering the use of the discriminant, students and mathematicians alike can gain a deeper understanding of quadratic equations and their applications in various fields.

Final Answer: B. one real-number solution