Find Zeros Of Horse Stall Function

Jan 4, 2026 by Andrew McMorgan 35 views

Finding the Zeros of the Horse Stall Function: A Math Guide

Hey guys! Ever wondered how math can model real-world scenarios? Today, we're diving into a fun problem from the world of county fairs and trusty steeds. We've got a function that tells us the number of horses occupying stalls at the county fairgrounds, and it's modeled by $h(x)=-3 x^2+30 x+225$ , where $x$ is the number of days since the first of June. Our mission, should we choose to accept it, is to rewrite this function in a form that reveals its zeros. Understanding the zeros of a function is super important because it tells us when the function's value is zero. In this context, it would mean finding out on which days there are no horses in the stalls, which is pretty interesting to think about, right? Let's get our math hats on and break this down step-by-step.

Understanding Quadratic Functions and Zeros

So, what exactly are we dealing with here? We have a quadratic function, $h(x)=-3 x^2+30 x+225$ . Quadratic functions are those awesome mathematical expressions that create a U-shaped or upside-down U-shaped graph, known as a parabola. The general form you usually see is $ax^2 + bx + c$ . In our case, $a = -3$ , $b = 30$ , and $c = 225$ . The 'zeros' of a function, guys, are the x-values where the function's output ( $h(x)$ in this case) equals zero. Graphically, these are the points where the parabola crosses the x-axis. For our horse stall problem, finding the zeros means finding the specific days ( $x$ ) when the number of horses ( $h(x)$ ) is zero. This could indicate the fair has ended, or perhaps a period where the stalls are unexpectedly empty. Finding these zeros is crucial for understanding the full behavior of our model and when it predicts no horses. We're not just looking at the peak number of horses, but also the boundaries of when the model is even relevant or predicts an absence. It’s like finding the start and end points of a particular trend.

There are a few ways to find the zeros of a quadratic function. The most common method is using the quadratic formula, ax^2+bx+c $) into the factored form ($ a(x-p)(x-q) $) or vertex form ($ a(x-h)^2+k$). The factored form directly shows the zeros $p$ and $q$ . Let's try to factor the given function $h(x) = -3x^2 + 30x + 225$ .

Factoring the Quadratic Function

To rewrite $h(x) = -3x^2 + 30x + 225$ in a form that reveals the zeros, our best bet is to factor it. Factoring is like breaking down a number into its prime factors, but for polynomials. We want to express our quadratic as a product of linear terms. First, let's look for a common factor among all the terms. We have $-3x^2$ , $30x$ , and $225$ . The greatest common divisor (GCD) of the coefficients is 3. Also, all terms have a negative sign, so we can factor out $-3$ . Let's pull out $-3$ from the expression:

$h(x) = -3(x^2 - 10x - 75)$

Now, we need to factor the quadratic expression inside the parentheses: $x^2 - 10x - 75$ . We are looking for two numbers that multiply to $-75$ and add up to $-10$ . Let's list out pairs of factors for $-75$ :

1 and -75 (sum: -74)
-1 and 75 (sum: 74)
3 and -25 (sum: -22)
-3 and 25 (sum: 22)
5 and -15 (sum: -10)
-5 and 15 (sum: 10)

Bingo! The pair $-15$ and $5$ adds up to $-10$ . So, we can factor $x^2 - 10x - 75$ as $(x - 15)(x + 5)$ .

Putting it all together, our factored form of the function is:

$h(x) = -3(x - 15)(x + 5)$

This is exactly the form that reveals the zeros.

Identifying the Zeros

Now that we have the function in its factored form, $h(x) = -3(x - 15)(x + 5)$ , finding the zeros is a piece of cake! Remember, the zeros are the values of $x$ that make $h(x) = 0$ . For a product of terms to be zero, at least one of the terms must be zero. So, we set our factored expression equal to zero:

$-3(x - 15)(x + 5) = 0$

Since $-3$ is a constant and not zero, we can ignore it for finding the roots. We only need to consider the factors that contain $x$ :

$(x - 15)(x + 5) = 0$

This equation is true if either:

$x - 15 = 0 ext{ or } x + 5 = 0$

Solving the first equation:

$x - 15 = 0 ightarrow x = 15$

Solving the second equation:

$x + 5 = 0 ightarrow x = -5$

So, the zeros of the function are $x = 15$ and $x = -5$ . This means that according to our model, the number of horses in the stalls would be zero on day 15 and on day -5. Since $x$ represents the number of days since the first of June, a negative value for $x$ might refer to days before June 1st. Day 15 represents 15 days after June 1st, which is June 16th. Day -5 would represent 5 days before June 1st, which is May 27th. These are the key points where the number of horses is predicted to be zero by this specific quadratic model. It's super cool how factoring directly gives us these critical values!

The Significance of the Zeros in Context

Let's bring it back to our horse stall scenario. We found the zeros to be $x = 15$ and $x = -5$ . What does this actually mean for the county fair? The variable $x$ represents the number of days since the first of June. So, $x = 15$ corresponds to June 16th. This suggests that on June 16th, the model predicts there will be zero horses in the stalls. This could logically mark the end of the fair or a significant decrease in attendance. The other zero, $x = -5$ , corresponds to May 27th. This means that 5 days before June 1st, the model also predicts zero horses. This is interesting because it implies the model might be designed to cover a period slightly before the fair officially begins, or perhaps it's just a mathematical artifact of the quadratic equation. When we talk about the practical application of the zeros, we usually focus on the positive and relevant values. In this case, $x=15$ (June 16th) likely represents the day the fair concludes or when the stalls become empty. The negative zero might be less relevant for the actual fair period but is mathematically crucial for defining the parabola's intercepts. It's important to remember that mathematical models are simplifications of reality. This quadratic function $h(x)=-3 x^2+30 x+225$ might only be accurate for a certain range of days. For example, it might not make sense to have a negative number of horses, so values of $x$ that result in $h(x) < 0$ (and are not the zeros we found) might represent periods where the model is no longer applicable. Understanding these zeros helps us define the useful domain of our model. If the fair runs from, say, June 1st to June 15th, then $x=15$ might be right at the end, or slightly after, indicating the stalls are cleared by then. The negative zero, $x=-5$ , is likely outside the typical operational period of a fair, so its practical meaning might be limited. However, knowing these points allows us to interpret the overall trend and predict when the stall occupancy hits zero.

Conclusion: Unlocking the Function's Secrets

So there you have it, math enthusiasts! We took the function $h(x)=-3 x^2+30 x+225$ , which models the number of horses in stalls, and successfully rewrote it in factored form: $h(x) = -3(x - 15)(x + 5)$ . This transformation is super handy because it directly reveals the zeros of the function, which we found to be $x = 15$ and $x = -5$ . In the context of our problem, these zeros tell us the specific days when the number of horses is predicted to be zero. While the negative zero might be less practically significant for the fair itself, the positive zero ( $x=15$ , or June 16th) likely signifies the day the stalls are emptied. Understanding how to convert a quadratic from its standard form to its factored form is a fundamental skill in algebra, and it allows us to quickly identify crucial points on the graph, like the x-intercepts (the zeros!). It helps us interpret the model's behavior and its limitations. Keep practicing these techniques, and you'll be factoring like a pro in no time! Math is all about understanding patterns and relationships, and finding the zeros is a key step in unraveling the story that a function is trying to tell us. Pretty neat, huh?