Finding M Values For No X-Intercepts In Y=mx^2-5x-2
Introduction: Understanding Quadratic Functions and x-Intercepts
In the realm of mathematics, quadratic functions play a pivotal role, particularly in algebra and calculus. These functions, characterized by the general form y = ax² + bx + c, produce a parabolic graph when plotted on a coordinate plane. The x-intercepts of this parabola, also known as the roots or zeros of the quadratic equation, are the points where the graph intersects the x-axis. These points hold significant information about the function's behavior and solutions. Determining the x-intercepts involves finding the values of x for which y equals zero. This is commonly achieved by solving the quadratic equation ax² + bx + c = 0. The solutions can be real, representing actual intersections with the x-axis, or complex, indicating that the parabola does not intersect the x-axis.
Understanding the conditions under which a quadratic function has no x-intercepts is crucial in various mathematical contexts. This occurs when the solutions to the quadratic equation are complex numbers. The nature of the solutions is dictated by the discriminant, a key component of the quadratic formula. The discriminant, denoted as Δ, is given by the formula Δ = b² - 4ac. If Δ is positive, the equation has two distinct real roots, meaning the parabola intersects the x-axis at two points. If Δ is zero, the equation has one real root (a repeated root), and the parabola touches the x-axis at one point. However, if Δ is negative, the equation has no real roots, indicating that the parabola does not intersect the x-axis. This article delves into the specifics of finding the values of m for which the quadratic function y = mx² - 5x - 2 has no x-intercepts, providing a comprehensive explanation and solution.
Problem Statement: Finding the Range of m for No x-Intercepts
Our primary objective is to determine the specific values of m for which the quadratic function given by the equation y = mx² - 5x - 2 does not intersect the x-axis. In mathematical terms, we are looking for the range of m that ensures the graph of the quadratic function has no x-intercepts. This condition implies that the equation mx² - 5x - 2 = 0 has no real solutions. As discussed earlier, the absence of real solutions corresponds to a negative discriminant. Therefore, our task boils down to finding the values of m that make the discriminant of the given quadratic equation negative. This involves identifying the coefficients a, b, and c in the quadratic equation, calculating the discriminant using the formula Δ = b² - 4ac, and then setting up an inequality to solve for m when Δ < 0. By systematically following these steps, we can accurately determine the range of m that satisfies the condition of no x-intercepts.
Methodology: Applying the Discriminant Condition
To solve this problem, we will utilize the concept of the discriminant, which, as mentioned earlier, is a crucial factor in determining the nature of the roots of a quadratic equation. The discriminant (Δ) is defined as b² - 4ac, where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. In our specific case, the equation is mx² - 5x - 2 = 0, so we can identify the coefficients as a = m, b = -5, and c = -2. For the quadratic function to have no x-intercepts, the discriminant must be negative (Δ < 0). This condition ensures that the quadratic equation has no real solutions, and hence the graph does not intersect the x-axis. The process involves substituting the identified coefficients into the discriminant formula and setting up the inequality b² - 4ac < 0. Solving this inequality will yield the range of values for m that satisfy the given condition.
The steps we will follow are:
- Identify the Coefficients: Determine the values of a, b, and c from the quadratic equation mx² - 5x - 2 = 0.
- Calculate the Discriminant: Substitute the values of a, b, and c into the discriminant formula Δ = b² - 4ac.
- Set up the Inequality: Establish the inequality Δ < 0, representing the condition for no real roots.
- Solve for m: Solve the inequality to find the range of values for m that satisfy the condition.
By carefully executing these steps, we can arrive at the correct solution for the range of m for which the given quadratic function has no x-intercepts. This methodical approach ensures accuracy and clarity in our solution.
Detailed Solution: Step-by-Step Calculation
Let's proceed with the solution step by step, ensuring clarity and precision in our calculations. First, we identify the coefficients of the quadratic equation y = mx² - 5x - 2. Comparing this equation with the general form ax² + bx + c = 0, we find:
- a = m
- b = -5
- c = -2
Next, we calculate the discriminant (Δ) using the formula Δ = b² - 4ac. Substituting the values of a, b, and c, we get:
Δ = (-5)² - 4(m)(-2) Δ = 25 + 8m
For the quadratic function to have no x-intercepts, the discriminant must be negative. This means:
Δ < 0 25 + 8m < 0
Now, we solve this inequality for m. Subtracting 25 from both sides, we have:
8m < -25
Dividing both sides by 8, we get:
m < -25/8
Therefore, the condition for the graph of y = mx² - 5x - 2 to have no x-intercepts is m < -25/8. This result indicates that when m is less than -25/8, the quadratic equation mx² - 5x - 2 = 0 has no real roots, and the parabola represented by the function does not intersect the x-axis.
Conclusion: The Range of m for No x-Intercepts
In conclusion, we have successfully determined the range of values for m that ensure the graph of the quadratic function y = mx² - 5x - 2 has no x-intercepts. By applying the concept of the discriminant and solving the inequality b² - 4ac < 0, we found that the condition is m < -25/8. This means that when the coefficient m is less than -25/8, the parabola represented by the function y = mx² - 5x - 2 does not intersect the x-axis. This understanding is crucial in various mathematical applications, such as analyzing the behavior of quadratic functions, solving quadratic inequalities, and determining the nature of roots of quadratic equations.
The discriminant serves as a powerful tool in understanding the characteristics of quadratic functions. It allows us to predict whether a quadratic equation will have real solutions, and if so, how many. In the case of no x-intercepts, the discriminant being negative indicates that the parabola lies entirely above or below the x-axis, never crossing it. This concept is fundamental in algebra and calculus, providing a basis for more advanced topics such as complex numbers and conic sections. The problem-solving approach demonstrated here, involving the use of the discriminant, is applicable to a wide range of quadratic equation problems, making it an essential skill for any mathematics student or professional.
The answer to the question “For what values of m does the graph of y = mx² - 5x - 2 have no x-intercepts?” is:
B. m < -25/8
This result is derived from a systematic application of the discriminant condition, highlighting the importance of understanding and utilizing key mathematical concepts in problem-solving. The thorough explanation and step-by-step solution provided in this article offer a comprehensive guide for anyone seeking to master the intricacies of quadratic functions and their properties.
