Finding Zeros Of Quadratic Function F(x) = X^2 + 10x + 24

Finding the zeros of a quadratic function is a fundamental concept in algebra, with applications ranging from physics to engineering and economics. In this article, we will delve into the process of finding the zeros of a given quadratic function, $f(x) = x^2 + 10x + 24$. We will explore various methods, including factoring, using the quadratic formula, and completing the square. Understanding these methods will not only help in solving this particular problem but also equip you with the skills to tackle a wide range of quadratic equations.

Understanding Quadratic Functions and Zeros

Before we dive into solving the specific problem, let’s clarify what quadratic functions and their zeros are. A quadratic function is a polynomial function of degree two, which can be generally written in the form $f(x) = ax^2 + bx + c$, where a, b, and c are constants and $a ≠ 0$. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards (if a > 0) or downwards (if a < 0). Zeros, also known as roots or x-intercepts, are the values of x for which the function $f(x)$ equals zero. Graphically, these are the points where the parabola intersects the x-axis. A quadratic function can have two real zeros, one real zero (a repeated root), or no real zeros (two complex roots). Finding these zeros is crucial for understanding the behavior of the function and its applications.

To find the zeros, we need to solve the quadratic equation $ax^2 + bx + c = 0$. There are several methods to accomplish this, each with its own advantages and suitability depending on the specific equation. The most common methods include factoring, using the quadratic formula, and completing the square. Factoring is often the quickest method when the quadratic expression can be easily factored. The quadratic formula is a universal method that works for any quadratic equation, regardless of whether it can be factored easily. Completing the square is a method that can be used to derive the quadratic formula and is also useful in transforming the quadratic equation into a form that reveals the vertex of the parabola. Each of these methods provides a different perspective on solving quadratic equations and enhances our understanding of their properties.

Method 1: Factoring the Quadratic Function

In many cases, the easiest way to find the zeros of a quadratic function is by factoring. Factoring involves expressing the quadratic expression as a product of two binomials. This method is particularly effective when the coefficients are integers and the roots are rational numbers. To factor the quadratic function $f(x) = x^2 + 10x + 24$, we look for two numbers that multiply to the constant term (24) and add up to the coefficient of the linear term (10). Let’s consider the factors of 24: 1 and 24, 2 and 12, 3 and 8, 4 and 6. Among these pairs, 4 and 6 add up to 10, which is exactly what we need. Therefore, we can rewrite the quadratic expression as $(x + 4)(x + 6)$.

Now, to find the zeros, we set the factored expression equal to zero: $(x + 4)(x + 6) = 0$. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. This gives us two separate equations: $x + 4 = 0$ and $x + 6 = 0$. Solving these equations for x is straightforward. For the first equation, we subtract 4 from both sides, yielding $x = -4$. For the second equation, we subtract 6 from both sides, giving $x = -6$. Thus, the zeros of the quadratic function $f(x) = x^2 + 10x + 24$ are $x = -4$ and $x = -6$. Factoring provides a clear and direct way to find the zeros when the quadratic expression is factorable, making it an essential technique in solving quadratic equations.

Method 2: Using the Quadratic Formula

When factoring is not straightforward or the quadratic expression is not easily factorable, the quadratic formula provides a reliable method for finding the zeros. The quadratic formula is derived from the process of completing the square and can be applied to any quadratic equation in the form $ax^2 + bx + c = 0$. The formula is given by: $x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}$. In this formula, a, b, and c are the coefficients of the quadratic, linear, and constant terms, respectively. For our given function, $f(x) = x^2 + 10x + 24$, we have a = 1, b = 10, and c = 24. Substituting these values into the quadratic formula, we get:

x = \frac{-10 ± \sqrt{10^2 - 4(1)(24)}}{2(1)}$. Let's break this down step by step. First, we calculate the discriminant, which is the expression inside the square root: $b^2 - 4ac = 10^2 - 4(1)(24) = 100 - 96 = 4$. The discriminant is a critical part of the quadratic formula because it tells us about the nature of the roots. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root (a repeated root); and if it is negative, there are two complex roots. In our case, the discriminant is 4, which is positive, indicating that we have two distinct real roots. Now we can continue with the quadratic formula: $x = \frac{-10 ± \sqrt{4}}{2} = \frac{-10 ± 2}{2}$. This gives us two possible solutions for *x*. The first solution is $x = \frac{-10 + 2}{2} = \frac{-8}{2} = -4$. The second solution is $x = \frac{-10 - 2}{2} = \frac{-12}{2} = -6$. Therefore, using the quadratic formula, we find the zeros of the function $f(x) = x^2 + 10x + 24$ to be $x = -4$ and $x = -6$. The quadratic formula is a powerful tool that guarantees a solution for any quadratic equation, making it an indispensable technique in algebra. ## Method 3: Completing the Square Completing the square is another method for finding the zeros of a quadratic function. This method involves transforming the quadratic expression into a perfect square trinomial, which can then be easily solved. While it may seem more complex than factoring or using the quadratic formula, completing the square provides valuable insights into the structure of quadratic equations and is particularly useful in calculus and other advanced mathematical topics. To complete the square for the function $f(x) = x^2 + 10x + 24$, we first focus on the quadratic and linear terms, $x^2 + 10x$. We want to add and subtract a value that will make this expression a perfect square. The value we need to add and subtract is $\left(\frac{b}{2} ight)^2$, where *b* is the coefficient of the linear term. In our case, *b* = 10, so we have $\left(\frac{10}{2} ight)^2 = 5^2 = 25$. Thus, we add and subtract 25 within the expression: $f(x) = x^2 + 10x + 25 - 25 + 24$. Now, we can rewrite the first three terms as a perfect square: $f(x) = (x + 5)^2 - 25 + 24$. Simplifying further, we get $f(x) = (x + 5)^2 - 1$. To find the zeros, we set $f(x)$ equal to zero: $(x + 5)^2 - 1 = 0$. Next, we isolate the squared term by adding 1 to both sides: $(x + 5)^2 = 1$. Taking the square root of both sides, we get $x + 5 = ±\sqrt{1}$, which simplifies to $x + 5 = ±1$. This gives us two equations to solve: $x + 5 = 1$ and $x + 5 = -1$. Solving the first equation, we subtract 5 from both sides, yielding $x = 1 - 5 = -4$. Solving the second equation, we subtract 5 from both sides, giving $x = -1 - 5 = -6$. Therefore, the zeros of the quadratic function $f(x) = x^2 + 10x + 24$ found by completing the square are $x = -4$ and $x = -6$. Completing the square not only helps in finding the zeros but also transforms the quadratic function into vertex form, which reveals the vertex of the parabola. ## Conclusion and Answer In this article, we explored various methods to find the zeros of the quadratic function $f(x) = x^2 + 10x + 24$. We successfully used factoring, the quadratic formula, and completing the square to arrive at the same solutions. Each method provides a unique approach to solving quadratic equations, enhancing our understanding of their properties and applications. By factoring the quadratic expression, we found that $(x + 4)(x + 6) = 0$, which gave us the zeros $x = -4$ and $x = -6$. Using the quadratic formula, we substituted the coefficients into the formula $x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}$ and obtained the same zeros, $x = -4$ and $x = -6$. Completing the square involved transforming the quadratic expression into the form $(x + 5)^2 - 1 = 0$, which also led to the zeros $x = -4$ and $x = -6$. Therefore, the zeros of the quadratic function $f(x) = x^2 + 10x + 24$ are $x = -4$ and $x = -6$. Looking at the given options, the correct answer is: **D. $x = -4$ $x = -6$** Understanding how to find the zeros of quadratic functions is essential in algebra and has wide-ranging applications in various fields. Whether you prefer factoring, using the quadratic formula, or completing the square, having these tools at your disposal will enable you to solve a variety of quadratic equations efficiently and accurately. These methods are not just mathematical techniques; they are problem-solving strategies that can be applied in many different contexts.