Find The Zeros Of F(x) = X + 5 - 2x^2

Hey there, math enthusiasts! Today, we're diving deep into the world of functions and tackling a common, yet super important, concept: finding the zeros of a function. Specifically, we're going to break down how to find the zeros of the function $f(x) = x + 5 - 2x^2$. You might have seen this problem pop up in your studies, and honestly, it's all about understanding what those 'zeros' actually represent and how to use the tools we have to find them. When we talk about the zeros of a function, we're essentially looking for the values of $x$ that make the function's output, $f(x)$, equal to zero. Think of it as finding where the graph of the function crosses the x-axis. These points are crucial for understanding the behavior of the function, whether you're analyzing its roots, sketching its graph, or solving related equations. So, let's get started on this journey to uncover those elusive zeros!

Understanding Quadratic Functions and Their Zeros

Alright guys, before we jump headfirst into solving for the zeros of $f(x) = x + 5 - 2x^2$, let's get a solid grip on what we're dealing with. This function, $f(x) = x + 5 - 2x^2$, is what we call a quadratic function. You can spot it a mile away because it has that $x^2$ term – the highest power of $x$ is two. Quadratic functions have a distinctive U-shape when graphed, which can either open upwards or downwards, depending on the sign of the coefficient of the $x^2$ term. In our case, the coefficient is -2, which means our parabola will be opening downwards. Now, the zeros of a function are the values of $x$ for which $f(x) = 0$. For a quadratic function, these are the points where the parabola intersects the x-axis. They are also commonly referred to as the roots of the quadratic equation. Finding these zeros is fundamental in algebra and has applications in physics, engineering, economics, and pretty much everywhere you find curves that can be modeled by a quadratic equation. The number of real zeros a quadratic function can have is either zero, one (if the vertex touches the x-axis), or two (if it crosses the x-axis at two distinct points). To find these zeros, we typically set the function equal to zero and solve the resulting quadratic equation. There are several methods for this, including factoring, completing the square, and the most universally applicable method, the quadratic formula. Given that our function $f(x) = x + 5 - 2x^2$ isn't immediately obvious to factor, the quadratic formula is going to be our best friend here. It's a powerful tool that guarantees we can find the zeros for any quadratic equation in the standard form $ax^2 + bx + c = 0$. So, understanding these basics sets the stage for us to confidently tackle our specific problem.

Setting Up the Equation to Find the Zeros

Okay, so we know we need to find the values of $x$ that make $f(x) = 0$. Our function is $f(x) = x + 5 - 2x^2$. To find the zeros, we simply set $f(x)$ equal to zero:

$x + 5 - 2x^2 = 0$

Now, this equation is a bit jumbled compared to the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$. The standard form is super helpful because it clearly identifies the coefficients $a$, $b$, and $c$, which we need for the quadratic formula. So, let's rearrange our equation to match that standard form. We want the $x^2$ term first, then the $x$ term, and finally the constant term. This gives us:

$-2x^2 + x + 5 = 0$

See? It's much cleaner now. In this form, we can easily identify our coefficients:

• $a = -2$ (the coefficient of $x^2$)
• $b = 1$ (the coefficient of $x$)
• $c = 5$ (the constant term)

Having these coefficients clearly defined is absolutely essential for the next step, which is applying the quadratic formula. If you mix up these values, you'll end up with the wrong answer, so it's worth double-checking. Remember, the signs are crucial! That negative sign on the $a$ coefficient is important and must be included. Now that we have our equation in perfect standard form and our coefficients identified, we are all set to plug these values into the quadratic formula and solve for $x$.

Applying the Quadratic Formula: The Key to Unlocking the Zeros

Alright, guys, this is where the magic happens! We've got our quadratic equation neatly arranged as $-2x^2 + x + 5 = 0$, and we've identified our coefficients: $a = -2$, $b = 1$, and $c = 5$. The quadratic formula is the ultimate tool for finding the zeros (or roots) of any quadratic equation in the form $ax^2 + bx + c = 0$. The formula itself is:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

This formula looks a bit intimidating at first glance, but it's just a systematic way to plug in our numbers and get our answers. Let's break it down and substitute our values for $a$, $b$, and $c$ step-by-step.

First, let's calculate the part under the square root, which is called the discriminant ($b^2 - 4ac$). This tells us about the nature of the roots, but we'll get to that.

Substitute $a = -2$, $b = 1$, and $c = 5$ into the discriminant:

$b^2 - 4ac = (1)^2 - 4(-2)(5)$

$= 1 - (-40)$

$= 1 + 40$

$= 41$

So, the discriminant is 41. Since 41 is a positive number, we know we're going to have two distinct real zeros, which is great! Now, let's plug this back into the full quadratic formula:

$$x = \frac{-(1) \pm \sqrt{41}}{2(-2)}$$

Now, we simplify the denominator and the term outside the square root:

-b = -(1) = -1$ $2a = 2(-2) = -4$ So, the formula becomes: $x = \frac{-1 \pm \sqrt{41}}{-4}

And there you have it! We've successfully applied the quadratic formula. The '$\pm$' symbol indicates that there are two possible solutions, one using the plus sign and one using the minus sign. These are the zeros of our function $f(x) = x + 5 - 2x^2$. It's super satisfying to see how the formula systematically leads us to the answer. Remember to always be careful with your signs during substitution and calculation – that's usually where most mistakes happen, guys!

Interpreting the Solutions and Matching with Options

We've arrived at our solutions using the quadratic formula: $x = \frac{-1 \pm \sqrt{41}}{-4}$. Now, let's talk about what this means and how to match it with the multiple-choice options provided. The '$\\pm$' symbol is key here; it tells us we have two distinct values for $x$:

1. Using the '+' sign: $x_1 = \frac{-1 + \sqrt{41}}{-4}$

2. Using the '-' sign: $x_2 = \frac{-1 - \sqrt{41}}{-4}$

These are the exact values of the zeros for the function $f(x) = x + 5 - 2x^2$. Now, let's look at the given options:

A. x=rac{-1 pm \sqrt{41}}{-4} B. x=rac{1 pm \sqrt{41}}{-4} C. x=rac{-1 pm \sqrt{39}}{-4} D. x=rac{1 pm \sqrt{39}}{-4}

By comparing our derived solution, $x = \frac{-1 \pm \sqrt{41}}{-4}$, directly with the options, we can see that Option A matches perfectly. It includes the $-1$ term in the numerator, the $\\pm$ symbol, the $\\sqrt{41}$ under the radical, and the $-4$ in the denominator. It's important to note that sometimes the options might be presented in a slightly different, but mathematically equivalent, form. For instance, you could multiply the numerator and denominator by -1 to get $x = \frac{1 \mp \sqrt{41}}{4}$, which represents the same two roots. However, in this case, Option A is an exact match. This step of comparing your calculated answer with the provided options is super critical in multiple-choice tests. Always double-check each component – the signs, the numbers, the radicals – to ensure a confident selection. You've successfully navigated the process of finding zeros for a quadratic function, which is a major win!

Conclusion: Mastering Quadratic Zeros

So there you have it, folks! We've successfully tackled the problem of finding the zeros for the quadratic function $f(x) = x + 5 - 2x^2$. By understanding that zeros are the $x$-values where $f(x) = 0$, and by strategically rearranging our function into the standard quadratic form $ax^2 + bx + c = 0$, we were able to identify our coefficients $a = -2$, $b = 1$, and $c = 5$. The trusty quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, served as our guide. Through careful substitution and simplification, we arrived at the solutions $x = \frac{-1 \pm \sqrt{41}}{-4}$. This clearly matched Option A among the given choices. Mastering the process of finding zeros is a fundamental skill in mathematics, opening doors to understanding graphical behavior, solving equations, and applying mathematical models to real-world problems. Remember, the key steps involve recognizing the type of function, setting it to zero, putting it in standard form, and applying the appropriate formula or method. Don't be intimidated by the formulas; break them down, substitute carefully, and always double-check your work, especially those pesky signs! Keep practicing, and you'll be finding the zeros of any function like a pro in no time. Happy problem-solving!