Determining The Value Of 12x-x²-32 When X Is Greater Than 8

Hey guys! Let's dive into a fun math problem today. We're going to figure out what happens to the expression 12x - x² - 32 when x is bigger than 8. This might sound a bit tricky at first, but trust me, we'll break it down step by step so it's super easy to understand. We'll use some basic algebra and a bit of logical thinking to get to the bottom of this. So, grab your thinking caps, and let's get started!

Understanding the Expression: 12x - x² - 32

To understand the behavior of the quadratic expression 12x - x² - 32 when x is greater than 8, let's first rearrange it into a more familiar form. We can rewrite the expression as -x² + 12x - 32. This is a quadratic expression in the standard form of ax² + bx + c, where a = -1, b = 12, and c = -32. Recognizing this form is crucial because it allows us to use various techniques to analyze the expression, such as finding its roots, vertex, and understanding its overall shape. Because the coefficient of the x² term is negative (a = -1), we know that the parabola opens downwards. This means the graph of the expression will have a maximum point (a vertex) and the values of the expression will decrease as we move away from this vertex in either direction. To get a clearer picture of what’s going on, we can find the roots of the quadratic equation. The roots are the values of x for which the expression equals zero. Setting -x² + 12x - 32 = 0 and solving for x will give us these roots. Factoring the quadratic expression is one way to find the roots. We need to find two numbers that multiply to give -32 and add up to 12. Alternatively, we could use the quadratic formula, which is a general method for finding the roots of any quadratic equation. Understanding the roots will help us determine the intervals where the expression is positive, negative, or zero. Finally, understanding the components of the expression sets the stage for a more detailed analysis. We can explore how the expression changes as x varies, particularly when x is greater than 8, which is the main focus of our discussion. By understanding the behavior of quadratic expressions generally, we can apply this knowledge to our specific problem and gain valuable insights.

Finding the Roots of the Quadratic Equation

Finding the roots of the quadratic equation -x² + 12x - 32 = 0 is a key step in understanding the behavior of the expression. The roots are the values of x that make the expression equal to zero, and they provide critical points for analyzing the expression's sign (positive, negative, or zero). There are a couple of methods we can use to find these roots: factoring and using the quadratic formula. Let's start with factoring, as it can be a quicker method if the quadratic expression is easily factorable. Factoring involves rewriting the quadratic expression as a product of two binomials. In our case, we need to find two numbers that multiply to -32 and add up to 12. Those numbers are 4 and 8. Thus, we can rewrite the equation as -(x - 4)(x - 8) = 0. Setting each factor equal to zero gives us the roots: x - 4 = 0 implies x = 4, and x - 8 = 0 implies x = 8. So, the roots of the quadratic equation are x = 4 and x = 8. Alternatively, we can use the quadratic formula, which is a general method for finding the roots of any quadratic equation of the form ax² + bx + c = 0. The formula is given by x = (-b ± √(b² - 4ac)) / (2a). In our case, a = -1, b = 12, and c = -32. Plugging these values into the formula, we get x = (-12 ± √(12² - 4(-1)(-32))) / (2(-1)). Simplifying this, we have x = (-12 ± √(144 - 128)) / (-2), which further simplifies to x = (-12 ± √16) / (-2). Thus, x = (-12 ± 4) / (-2). This gives us two possible solutions: x = (-12 + 4) / (-2) = -8 / -2 = 4 and x = (-12 - 4) / (-2) = -16 / -2 = 8. Both methods confirm that the roots of the equation are x = 4 and x = 8. These roots are crucial because they divide the number line into intervals where the expression 12x - x² - 32 has a consistent sign (either positive or negative).

Analyzing the Expression When x > 8

Now that we've found the roots of the quadratic equation, we can analyze the behavior of the expression 12x - x² - 32 when x is greater than 8. The roots, x = 4 and x = 8, divide the number line into three intervals: x < 4, 4 < x < 8, and x > 8. We are specifically interested in the interval where x > 8. To determine the sign of the expression in this interval, we can pick a test value greater than 8 and plug it into the expression. Let's choose x = 9 as our test value. Substituting x = 9 into the expression 12x - x² - 32, we get 12(9) - (9)² - 32 = 108 - 81 - 32 = 108 - 113 = -5. The result is negative, which means that the expression 12x - x² - 32 is negative when x = 9. Since the quadratic expression changes sign only at its roots, the expression will be negative for all values of x greater than 8. Another way to think about this is to consider the factored form of the expression: -(x - 4)(x - 8). When x > 8, both factors (x - 4) and (x - 8) are positive. However, because of the negative sign in front of the factored expression, the entire expression becomes negative. For example, if x = 9, then (x - 4) = 5 and (x - 8) = 1, so -(x - 4)(x - 8) = -(5)(1) = -5, which is negative. This confirms our earlier finding. We can also think about the graph of the quadratic function y = -x² + 12x - 32. As we discussed earlier, the parabola opens downwards (because the coefficient of x² is negative) and has roots at x = 4 and x = 8. The part of the parabola where x > 8 lies below the x-axis, which means the y-values (i.e., the values of the expression) are negative. Therefore, we can confidently conclude that the expression 12x - x² - 32 is negative when x is greater than 8. This understanding is crucial for various applications, such as solving inequalities and analyzing mathematical models.

Conclusion: The Value is Negative

In conclusion, after analyzing the quadratic expression 12x - x² - 32 and determining its behavior when x is greater than 8, we've found that the value of the expression is always negative in this interval. We started by rearranging the expression into the standard quadratic form and identified the coefficients. This allowed us to understand that the parabola opens downwards due to the negative coefficient of the x² term. Then, we found the roots of the equation -x² + 12x - 32 = 0 using both factoring and the quadratic formula. The roots, x = 4 and x = 8, are critical points because they divide the number line into intervals where the expression maintains a consistent sign. By testing a value greater than 8 (we used x = 9) in the original expression, we found that the result was negative. This, combined with our understanding of the parabola's shape and the roots, confirmed that the expression 12x - x² - 32 is negative for all x > 8. We also considered the factored form of the expression, -(x - 4)(x - 8), and showed that when x > 8, both factors (x - 4) and (x - 8) are positive, but the negative sign in front makes the entire expression negative. This comprehensive analysis, using both algebraic techniques and graphical intuition, gives us a solid understanding of the expression's behavior. So, to reiterate, whenever x is greater than 8, the value of 12x - x² - 32 will always be negative. This type of analysis is fundamental in various mathematical contexts, including calculus, optimization problems, and real-world applications where quadratic relationships are involved. Understanding how expressions behave in different intervals is a valuable skill in problem-solving and mathematical reasoning.