Finding The Maximum Value Of M For H(m) = 0

In this article, we'll dive into a fascinating problem involving a function defined by a square root expression and explore how to determine the maximum possible value that satisfies a given condition. Specifically, we'll be working with the function h(x) = -√(x² + bx + c), where b and c are constants. Our goal is to find the greatest possible value of m such that h(m) = 0, given that the graph of y = h(x) passes through the points (2, 0) and (0, -√266). This problem combines concepts from algebra and coordinate geometry, offering a rich opportunity to apply our problem-solving skills.

Understanding the Function h(x) = -√(x² + bx + c)

The function h(x) = -√(x² + bx + c) is a variation of a square root function, modified by a negative sign and a quadratic expression inside the square root. Let's break down the key components to gain a clearer understanding:

• Square Root Function: The basic square root function, √x, only produces real values for non-negative inputs (x ≥ 0). This means the expression inside the square root, x² + bx + c, must be greater than or equal to zero for h(x) to be defined in the real number system. This condition will be crucial in determining the possible values of x. It's important to remember that the square root function always returns a non-negative value. Therefore, the negative sign in front of the square root in h(x) ensures that the function's output will always be non-positive (zero or negative). This implies that the graph of y = h(x) will lie on or below the x-axis. The quadratic expression inside the square root, x² + bx + c, is a parabola. The roots of this quadratic (where x² + bx + c = 0) will play a significant role in determining the domain of h(x) and the points where the graph intersects the x-axis. We can use the points given, (2,0) and (0,-√266), to find the values of b and c, which will further define the function. By understanding these individual components, we can start to visualize the behavior of h(x) and strategize how to find the greatest possible value of m where h(m) = 0. This involves setting up equations based on the given points and solving for the unknowns.

Using the Given Points (2, 0) and (0, -√266)

The fact that the graph of y = h(x) contains the points (2, 0) and (0, -√266) provides us with valuable information. Each point gives us a pair of x and y values that satisfy the equation h(x) = -√(x² + bx + c). We can use these points to create a system of equations and solve for the unknown constants, b and c. This is a standard technique in algebra – using known points on a curve to determine the equation of the curve. Substituting x = 2 and h(2) = 0 into the equation, we get:

*0 = -√(2² + 2b + c)*

Since the square root of a number is zero only when the number itself is zero, this simplifies to:

*4 + 2b + c = 0* (Equation 1)

Next, we substitute x = 0 and h(0) = -√266 into the equation:

*-√266 = -√(0² + 0b + c)*

This simplifies to:

*√266 = √c*

Squaring both sides, we find:

*c = 266* (Equation 2)

Now we have the value of c, and we can substitute it back into Equation 1 to solve for b:

*4 + 2b + 266 = 0*

*2b = -270*

*b = -135*

Therefore, we have found the values b = -135 and c = 266. This allows us to define the function h(x) completely, which is a crucial step in solving the problem. With b and c known, we can now focus on finding the values of m that satisfy h(m) = 0, and then determine the greatest possible value among them. This involves solving a quadratic equation, which we will address in the next section.

Finding m when h(m) = 0

Now that we know the function is defined as h(x) = -√(x² - 135x + 266), our next step is to find the values of m for which h(m) = 0. This means we need to solve the equation:

*0 = -√(m² - 135m + 266)*

As before, the square root is zero only when the expression inside is zero. Thus, we need to solve the quadratic equation:

*m² - 135m + 266 = 0*

This is a standard quadratic equation in the form am² + bm + c = 0, where a = 1, b = -135, and c = 266. We can solve this equation using the quadratic formula:

*m = (-b ± √(b² - 4ac)) / (2a)*

Substituting the values, we get:

*m = (135 ± √((-135)² - 4 * 1 * 266)) / (2 * 1)*

*m = (135 ± √(18225 - 1064)) / 2*

*m = (135 ± √17161) / 2*

*m = (135 ± 131) / 2*

This gives us two possible values for m:

*m₁ = (135 + 131) / 2 = 266 / 2 = 133*

*m₂ = (135 - 131) / 2 = 4 / 2 = 2*

So, the two values of m that make h(m) = 0 are m = 133 and m = 2. The question asks for the greatest possible value of m, which in this case is 133. It's important to note that we obtained two solutions, and we had to identify the larger one to answer the specific question asked. This highlights the importance of careful reading and understanding the question's requirements. We have now successfully found the greatest possible value of m that satisfies the given conditions.

Determining the Greatest Possible Value of m

After solving the quadratic equation, we found two possible values for m: m = 133 and m = 2. The question explicitly asks for the greatest possible value of m such that h(m) = 0. Comparing the two values, it's clear that 133 is greater than 2. Therefore, the greatest possible value of m is 133. This is the final answer to the problem. It’s crucial to always double-check what the question is asking for, as sometimes there might be multiple solutions, but only one that satisfies the specific criteria. In this case, while we found two values for m that make h(m) = 0, the question focused on the maximum possible value. This step emphasizes the importance of careful interpretation and attention to detail in problem-solving. We have now successfully navigated through the entire problem, from understanding the function and using the given points to finding the solutions and identifying the greatest possible value. This demonstrates a strong understanding of algebraic concepts and problem-solving techniques. This complete solution showcases the process of using given information to derive equations, solving those equations, and then applying the results to answer a specific question. Each step builds upon the previous one, highlighting the interconnectedness of mathematical concepts.

Conclusion

In summary, we were given the function h(x) = -√(x² + bx + c) and two points on its graph, (2, 0) and (0, -√266). We were tasked with finding the greatest possible value of m such that h(m) = 0. By substituting the given points into the function, we formed a system of equations and solved for the constants b and c. This allowed us to define the function as h(x) = -√(x² - 135x + 266). We then set h(m) = 0 and solved the resulting quadratic equation to find two possible values for m: 133 and 2. Finally, we identified the greatest of these values, which is 133. This problem required a combination of algebraic manipulation, equation solving, and careful interpretation of the question. The process involved several key steps, including substituting known values to form equations, solving a quadratic equation using the quadratic formula, and comparing multiple solutions to find the one that meets the specific criteria of the problem. The final answer, 133, represents the maximum value of m for which the function h(m) equals zero, given the constraints provided. This type of problem is representative of the challenges encountered in algebra and precalculus courses, emphasizing the importance of a strong foundation in algebraic principles and problem-solving strategies. For further exploration of functions and their properties, you might find resources on websites like Khan Academy to be beneficial.