Finding Zeros Of F(x) = X^2 + 8x + 4 In Simplest Radical Form
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In mathematics, determining the zeros of a function is a fundamental task, especially when dealing with polynomial functions. The zeros, also known as roots or x-intercepts, are the values of x for which the function f(x) equals zero. This article delves into the process of finding the zeros of the quadratic function $f(x) = x^2 + 8x + 4$, expressing the solutions in simplest radical form. We will explore the quadratic formula, a powerful tool for solving quadratic equations, and demonstrate its application to this specific function. Understanding how to find zeros is crucial in various mathematical contexts, including graphing functions, solving equations, and analyzing real-world problems modeled by quadratic relationships. This comprehensive guide will walk you through each step, ensuring a clear and thorough understanding of the method and the underlying concepts.
Understanding Quadratic Functions and Zeros
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Before diving into the solution, it's essential to understand what quadratic functions and their zeros represent. A quadratic function is a polynomial function of degree two, generally expressed in the form $f(x) = ax^2 + bx + c$, where a, b, and c are constants and $a \ne 0$. The zeros of a quadratic function are the values of x that make the function equal to zero, i.e., the solutions to the equation $ax^2 + bx + c = 0$. Graphically, these zeros correspond to the points where the parabola (the graph of the quadratic function) intersects the x-axis. A quadratic function can have two real zeros, one real zero (a repeated root), or two complex zeros, depending on the discriminant ($b^2 - 4ac$) of the quadratic equation. If the discriminant is positive, there are two distinct real roots; if it's zero, there is one real root (a repeated root); and if it's negative, there are two complex roots. Understanding this relationship between the discriminant and the nature of the roots is crucial for solving quadratic equations effectively. This foundational knowledge sets the stage for applying the quadratic formula to find the zeros of the given function.
The Quadratic Formula: A Key to Finding Zeros
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The quadratic formula is a cornerstone in solving quadratic equations. It provides a direct method for finding the zeros of any quadratic function, regardless of whether the equation can be easily factored. The formula is derived from the process of completing the square and is given by:$x = rac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this formula, a, b, and c are the coefficients of the quadratic equation $ax^2 + bx + c = 0$. The $\pm$ symbol indicates that there are generally two solutions, one obtained by adding the square root term and the other by subtracting it. The expression inside the square root, $b^2 - 4ac$, is known as the discriminant, as mentioned earlier, and it determines the nature of the roots. The quadratic formula is a powerful tool because it works for all quadratic equations, even those that are difficult or impossible to factor by traditional methods. It transforms the problem of finding zeros into a straightforward process of substituting coefficients and simplifying the resulting expression. Mastering the quadratic formula is essential for anyone working with quadratic functions and equations.
Applying the Quadratic Formula to $f(x) = x^2 + 8x + 4$
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Now, let's apply the quadratic formula to the given function, $f(x) = x^2 + 8x + 4$. First, we identify the coefficients: a = 1, b = 8, and c = 4. Next, we substitute these values into the quadratic formula:$x = rac-8 \pm \sqrt{8^2 - 4(1)(4)}}{2(1)}$. This substitution is a critical step, ensuring that the correct values are placed in the appropriate positions within the formula. Now, we simplify the expression step by step. First, we calculate the square}2}$. Next, we subtract inside the square root}{2}$. This simplification process is crucial for arriving at the correct solution. The next step involves simplifying the radical, which will lead us to the simplest radical form of the zeros.
Simplifying the Radical: Expressing in Simplest Radical Form
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The next step in finding the zeros is to simplify the radical $\sqrt48}$. To do this, we look for the largest perfect square that divides 48. The largest perfect square factor of 48 is 16, since $48 = 16 \times 3$. Therefore, we can rewrite $\sqrt{48}$ as $\sqrt{16 \times 3}$. Using the property of square roots that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, we can further simplify this to $\sqrt{16} \times \sqrt{3}$. Since $\sqrt{16} = 4$, we have $\sqrt{48} = 4\sqrt{3}$. Substituting this simplified radical back into our expression for x, we get}{2}$. Now, we can simplify the entire expression by dividing both the numerator and the denominator by their greatest common factor, which is 2. This simplification process is essential for expressing the zeros in their simplest radical form, which is a key requirement of the problem. By breaking down the radical and simplifying the fraction, we ensure that our final answer is both accurate and in the most concise form.
Final Simplification and the Zeros of the Function
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To complete the process, we simplify the expression $x = rac-8 \pm 4\sqrt{3}}{2}$. We divide both the numerator and the denominator by 22} \pm rac{4\sqrt{3}}{2}$. This simplifies to$. Therefore, the zeros of the function $f(x) = x^2 + 8x + 4$ expressed in simplest radical form are $x = -4 + 2\sqrt{3}$ and $x = -4 - 2\sqrt{3}$. This result corresponds to option A. These zeros represent the x-intercepts of the parabola defined by the quadratic function. The ability to find and express zeros in simplest radical form is a crucial skill in algebra and calculus, with applications in various fields such as physics, engineering, and economics. Understanding the steps involved in applying the quadratic formula and simplifying radicals ensures accurate and efficient problem-solving. This final simplification provides a clear and concise solution to the original problem, highlighting the importance of each step in the process.
Conclusion
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In conclusion, we have successfully found the zeros of the quadratic function $f(x) = x^2 + 8x + 4$ and expressed them in simplest radical form. By applying the quadratic formula, simplifying the radical, and performing the necessary arithmetic operations, we determined that the zeros are $x = -4 \pm 2\sqrt{3}$. This process demonstrates the power and utility of the quadratic formula in solving quadratic equations. Understanding how to find zeros is not only a fundamental skill in mathematics but also a valuable tool for solving real-world problems that can be modeled by quadratic functions. Mastering these techniques provides a solid foundation for further studies in algebra, calculus, and related fields. The ability to confidently and accurately find zeros in simplest radical form is a testament to a strong understanding of algebraic principles and problem-solving strategies.
Therefore, the correct answer is A. $x = -4 \pm 2\sqrt{3}$
