Find The X-intercepts Of H(x) = 3x^2 - 75

Understanding how to find the x-intercepts of a function is a fundamental skill in mathematics, particularly when analyzing quadratic equations like the one presented: $h(x) = 3x^2 - 75$. The x-intercepts, also known as the roots or zeros of a function, are the points where the graph of the function crosses or touches the x-axis. At these points, the y-value (or the function's output, $h(x)$ in this case) is always zero. Therefore, to find the x-intercepts, we set $h(x)$ equal to 0 and solve for $x$. This process involves algebraic manipulation and an understanding of quadratic equations. For the given function, $h(x) = 3x^2 - 75$, we are essentially looking for the values of $x$ that make the equation $3x^2 - 75 = 0$ true. This specific type of quadratic equation, which lacks a linear term (the $bx$ term in $ax^2 + bx + c$), can often be solved by isolating the $x^2$ term and then taking the square root of both sides. It's a straightforward method that doesn't require the more complex quadratic formula, although the quadratic formula would also yield the correct answer. The key is to correctly identify the structure of the equation and apply the appropriate mathematical techniques. This is crucial for graphing functions, understanding their behavior, and solving various real-world problems that can be modeled by quadratic relationships, such as projectile motion, optimization problems, and even in economics to find break-even points. Mastering this skill will build a strong foundation for more advanced algebraic and calculus concepts.

To begin solving for the x-intercepts of $h(x) = 3x^2 - 75$, our first step is to set the function equal to zero: $3x^2 - 75 = 0$. The objective now is to isolate the $x^2$ term. We can do this by adding 75 to both sides of the equation. This operation maintains the equality, transforming the equation into $3x^2 = 75$. Once we have $3x^2$ isolated, the next logical step is to get $x^2$ by itself. We achieve this by dividing both sides of the equation by 3. Performing this division, we get $x^2 = \frac{75}{3}$. Simplifying the fraction $\frac{75}{3}$ gives us $x^2 = 25$. Now, we have a simple equation where $x^2$ is equal to a constant. To find the value(s) of $x$, we need to take the square root of both sides of the equation. It is extremely important to remember that when taking the square root of both sides of an equation in this context, there are typically two possible solutions: a positive root and a negative root. This is because squaring both a positive number and its negative counterpart results in the same positive number. Therefore, taking the square root of 25 yields two values for $x$: $x = \sqrt{25}$ and $x = -\sqrt{25}$. Calculating these values, we find that $x = 5$ and $x = -5$. These are the x-intercepts of the function $h(x) = 3x^2 - 75$. They represent the points where the parabola defined by this function crosses the x-axis. These points are $(5, 0)$ and $(-5, 0)$ on the Cartesian plane. This method is efficient for quadratics in the form $ax^2 + c = 0$, where the linear term is absent, making the solution process direct and less prone to errors compared to applying the full quadratic formula.

Let's recap the process for finding the x-intercepts of $h(x) = 3x^2 - 75$ to ensure clarity and reinforce understanding. The core principle is that x-intercepts occur where the function's output is zero, meaning $h(x) = 0$. So, we begin by setting the given function equal to zero: $3x^2 - 75 = 0$. Our goal is to solve this equation for $x$. The first algebraic step is to isolate the term containing $x^2$. We accomplish this by adding 75 to both sides of the equation, resulting in $3x^2 = 75$. Next, to isolate $x^2$ itself, we divide both sides of the equation by the coefficient of $x^2$, which is 3. This operation yields $x^2 = \frac{75}{3}$. Simplifying the fraction on the right side, we find $x^2 = 25$. The final step to find $x$ is to take the square root of both sides. It is crucial to remember that taking the square root introduces two possible solutions: a positive and a negative one. Thus, $x = \pm\sqrt{25}$. Evaluating the square root of 25, we get $x = 5$ and $x = -5$. Therefore, the x-intercepts of the function $h(x) = 3x^2 - 75$ are $x = 5$ and $x = -5$. If, after following this procedure, we had reached a point where $x^2$ was equal to a negative number (e.g., $x^2 = -25$), and we were only considering real number solutions, then we would conclude that there are no real x-intercepts, and the answer would be DNE (Does Not Exist). However, in this specific case, we found two distinct real solutions. These intercepts are significant because they mark the points where the graph of the quadratic function intersects the horizontal axis. For this particular function, which is a parabola opening upwards and shifted vertically, these intercepts provide key insights into its position and behavior on the coordinate plane. A thorough grasp of this method allows for the accurate identification of these critical points for any quadratic function that can be manipulated into a similar form. This skill is foundational for further mathematical exploration, including the study of graphing functions and solving systems of equations.

To further solidify our understanding, let's consider the graphical interpretation of these x-intercepts. The function $h(x) = 3x^2 - 75$ represents a parabola. The coefficient of the $x^2$ term, which is 3, is positive, indicating that the parabola opens upwards. The constant term, -75, represents the y-intercept, meaning the graph crosses the y-axis at $(0, -75)$. The x-intercepts we calculated, $x = 5$ and $x = -5$, correspond to the points where this upward-opening parabola intersects the x-axis. These points are $(5, 0)$ and $(-5, 0)$. The vertex of the parabola is located at the midpoint between the x-intercepts, which in this case is $x = \frac{5 + (-5)}{2} = 0$. The y-coordinate of the vertex is $h(0) = 3(0)^2 - 75 = -75$. Thus, the vertex of the parabola is at $(0, -75)$, which is also its y-intercept. The symmetry of the parabola is evident, with the y-axis acting as the axis of symmetry. The distance from the axis of symmetry to each x-intercept is 5 units. This graphical perspective confirms our algebraic findings. If, hypothetically, the equation had led to $x^2 = -25$, it would imply that no real number squared equals -25. In such a scenario, the parabola would never touch or cross the x-axis, meaning there would be no real x-intercepts. This could happen if, for example, the function was $g(x) = 3x^2 + 75$. Setting $g(x) = 0$ would lead to $3x^2 = -75$, and $x^2 = -25$, which has no real solutions. In such cases, the correct answer for the x-intercepts would be 'DNE' (Does Not Exist). The ability to find x-intercepts is not just an academic exercise; it's vital in many applications. For instance, in physics, when modeling the trajectory of a projectile, the x-intercepts can represent the points where the object lands. In economics, they might signify the production levels at which a company breaks even. Therefore, mastering this skill provides a powerful tool for analyzing and solving a wide range of problems.

The Importance of Real Solutions

When we solve for the x-intercepts of a function like $h(x) = 3x^2 - 75$, we are typically interested in real number solutions unless the problem explicitly states otherwise (e.g., asking for complex roots). The process involves setting $h(x) = 0$, which leads to $3x^2 - 75 = 0$. By isolating $x^2$, we obtained $x^2 = 25$. Taking the square root of both sides gives us $x = \pm\sqrt{25}$. The square root of 25 is a real number, 5. Therefore, we get two distinct real x-intercepts: $x = 5$ and $x = -5$. These are valid solutions because they are real numbers. If, however, the equation had resulted in a situation like $x^2 = -9$, and we were looking for real solutions, we would have to conclude that there are no real numbers that satisfy this equation. The square of any real number is always non-negative (zero or positive). Consequently, if the algebraic steps lead to $x^2$ equaling a negative number, it means the function's graph does not intersect the x-axis in the real number plane. In such cases, the appropriate response, as specified in the prompt, is 'DNE' (Does Not Exist). This distinction between real and complex solutions is crucial in many areas of mathematics and science. For instance, when analyzing physical phenomena, we are almost always concerned with real-world measurements and quantities, which are represented by real numbers. Complex solutions, while mathematically valid, might not have a direct physical interpretation in many contexts. Therefore, when asked to find x-intercepts without further specification, the convention is to seek real roots. The function $h(x) = 3x^2 - 75$ is a good example of a quadratic function that yields real x-intercepts, showcasing a common scenario in algebra. The ability to identify when real solutions exist and when they do not is just as important as finding the solutions themselves.

Conclusion

In conclusion, finding the x-intercepts of the function $h(x) = 3x^2 - 75$ involves setting the function equal to zero and solving for $x$. This process led us to the equation $3x^2 - 75 = 0$, which we solved by isolating $x^2$ to get $x^2 = 25$. Taking the square root of both sides, we found two real solutions: $x = 5$ and $x = -5$. These are the x-intercepts of the function. It's important to remember that quadratic functions can have zero, one, or two distinct real x-intercepts. The number of intercepts depends on the discriminant of the quadratic equation or, in simpler cases like this one, on whether $x^2$ ends up being positive, zero, or negative after algebraic manipulation. If $x^2$ had been negative, the answer would be DNE. This skill is a cornerstone of understanding quadratic functions and their graphical representations. For more information on quadratic functions and their properties, you can refer to resources like Khan Academy's algebra section.