Finding Zeros Of Quadratic Functions In Radical Form

When we talk about the zeros of a function, especially a quadratic function like $f(x)=x^2+8 x+4$, we're essentially asking: "At what x-values does the function's output (y-value) equal zero?" In simpler terms, where does the graph of this parabola cross the x-axis? Finding these zeros is a fundamental concept in algebra, and often, the solutions aren't nice, neat whole numbers. This is where the idea of expressing them in simplest radical form comes into play. It's a way to represent irrational numbers precisely, rather than resorting to approximations. For our function $f(x)=x^2+8 x+4$, we need to solve the equation $x^2+8 x+4=0$ for $x$. Since this quadratic doesn't easily factor, we'll turn to a powerful tool: the quadratic formula. The quadratic formula is derived from completing the square on the general quadratic equation $ax^2+bx+c=0$, and it gives us the solutions for $x$ as x = rac{-b rac{plusmnrac{right paren rac{b^2-4ac}}{2a}. In our case, $a=1$, $b=8$, and $c=4$. Plugging these values into the formula, we get x = rac{-8 rac{plusmnrac{right paren rac{8^2-4(1)(4)}}{2(1)}}{}. This simplifies to x = rac{-8 rac{plusmnrac{right paren rac{64-16}}{2}}{}, which further reduces to x = rac{-8 rac{plusmnrac{right paren rac{48}}{2}}{}. Now, we need to express this in the simplest radical form. The term rac{right paren rac{48}}{} contains a radical that can be simplified. We look for the largest perfect square factor of 48. That would be 16, since $16 imes 3 = 48$. So, rac{right paren rac{48}}{} = rac{right paren rac{16 imes 3}}{} = rac{right paren rac{16}}{} imes rac{right paren rac{3}}{} = 4rac{right paren rac{3}}{}. Substituting this back into our solution, we have x = rac{-8 rac{plusmnrac{right paren rac{4 imes 3}}{2}}{}. Finally, we can divide both terms in the numerator by 2: x = rac{-8}{2} rac{plusmnrac{right paren rac{4rac{right paren rac{3}}{2}}}{}. This gives us our final answer in simplest radical form: x=-4 rac{plusmnrac{right paren rac{2rac{right paren rac{3}}{}}{}}{}. This detailed step-by-step process ensures that we not only find the zeros but also present them in their most accurate and simplified mathematical format. It's a crucial skill for tackling more complex algebraic problems and understanding the behavior of functions.

Understanding the Components of the Quadratic Formula

The quadratic formula is an indispensable tool for solving quadratic equations of the form $ax^2+bx+c=0$, especially when factoring becomes challenging or impossible with integers. Let's break down its components and how they apply to finding the zeros of $f(x)=x^2+8 x+4$. The formula itself is x = rac{-b rac{plusmnrac{right paren rac{b^2-4ac}}{2a}}. Here, $a$, $b$, and $c$ are the coefficients of the quadratic equation. In our specific problem, $a=1$ (the coefficient of $x^2$), $b=8$ (the coefficient of $x$), and $c=4$ (the constant term). The term under the square root, rac{right paren rac{b^2-4ac}}{}, is called the discriminant. The discriminant is incredibly important because it tells us about the nature of the roots (the zeros). If the discriminant is positive, there are two distinct real roots. If it's zero, there's exactly one real root (a repeated root). If it's negative, there are two complex conjugate roots. For $f(x)=x^2+8 x+4$, the discriminant is $8^2 - 4(1)(4) = 64 - 16 = 48$. Since 48 is positive, we know there will be two distinct real zeros. The rac{plusmnrac{right paren }{} symbol indicates that there are two possible solutions: one where we add the square root term and one where we subtract it. The denominator, $2a$, ensures that our solutions are scaled correctly. In our case, $2a = 2(1) = 2$. So, substituting our values into the formula gives us x = rac{-8 rac{plusmnrac{right paren rac{48}}{2}}{}. This is a valid form of the answer, but the question asks for the simplest radical form. This means we need to simplify the radical rac{right paren rac{48}}{}.

Simplifying Radicals for Precise Answers

Simplifying radicals is a crucial step in expressing the zeros of a function in their most precise form, especially when dealing with irrational numbers that arise from the quadratic formula. For the function $f(x)=x^2+8 x+4$, we arrived at the expression x = rac{-8 rac{plusmnrac{right paren rac{48}}{2}}{}. The radical part, rac{right paren rac{48}}{}, needs simplification. The goal is to extract any perfect square factors from under the radical sign. We list the perfect squares: $1, 4, 9, 16, 25, 36, 49, ext{...}$. We look for the largest perfect square that divides 48. Testing them: 4 divides 48 (48 rac{right paren rac{12}}{}), 9 does not, 16 divides 48 ($48 = 16 imes 3$). Since 16 is the largest perfect square factor of 48, we can rewrite rac{right paren rac{48}}{} as rac{right paren rac{16 imes 3}}{}. Using the property of radicals that rac{right paren rac{ab}}{} = rac{right paren rac{a}}{} imes rac{right paren rac{b}}{}, we get rac{right paren rac{16 imes 3}}{} = rac{right paren rac{16}}{} imes rac{right paren rac{3}}{}. Since rac{right paren rac{16}}{} = 4, the simplified radical is 4rac{right paren rac{3}}{}. Now, we substitute this back into our solution: x = rac{-8 rac{plusmnrac{right paren rac{4rac{right paren rac{3}}{}}{2}}{}. The final step in simplifying is to divide both terms in the numerator by the denominator, 2. This means we divide -8 by 2 and 4rac{right paren rac{3}}{} by 2. So, x = rac{-8}{2} rac{plusmnrac{right paren rac{4rac{right paren rac{3}}{}}{2}}{}, which simplifies to x = -4 rac{plusmnrac{right paren rac{2rac{right paren rac{3}}{}}{}}{}. This is the solution expressed in simplest radical form. It's important to note that this matches option A: x=-4 rac{plusmnrac{right paren rac{2rac{right paren rac{3}}{}}{}}{}. Option B, x=-4 rac{plusmnrac{right paren rac{48}}{}}{}, is correct before simplifying the radical. Option C, x=rac{-8 rac{plusmnrac{right paren rac{3}}{}}{2}}, incorrectly simplifies the radical part. Option D, x=rac{-4 rac{plusmnrac{right paren rac{4rac{right paren rac{3}}{}}{2}}}{}, has an incorrect simplification of the radical term. Therefore, understanding how to simplify radicals is key to arriving at the correct and most concise answer.

The Significance of Radical Form in Mathematics

Expressing the zeros of a function, particularly quadratic functions, in simplest radical form is more than just an academic exercise; it's about maintaining precision and understanding the nature of mathematical solutions. When we solve equations like $x^2+8 x+4=0$ and arrive at results involving square roots that cannot be simplified to rational numbers (like rac{right paren rac{3}}{}), radical form is the way to represent these exact values. Approximations, like using a decimal value for rac{right paren rac{3}}{} ext{ (approximately } 1.732), lose mathematical integrity when exactness is required. In fields like engineering, physics, and computer science, where calculations need to be precise, working with exact forms prevents the accumulation of errors. The process of simplifying radicals, as we did with rac{right paren rac{48}}{} to 4rac{right paren rac{3}}{}, is a standard procedure that ensures that the number under the radical sign has no perfect square factors other than 1. This makes the expression as compact and understandable as possible. When we then combine this with the other parts of the quadratic formula, x = rac{-8 rac{plusmnrac{right paren rac{48}}{2}}{}, and simplify further to x=-4 rac{plusmnrac{right paren rac{2rac{right paren rac{3}}{}}{}}{}, we have arrived at the most elegant and precise representation of the function's zeros. This form clearly shows that the zeros are symmetric around $x=-4$, with a distance of 2rac{right paren rac{3}}{} from that central point. Understanding this structure is vital for graphing the function and interpreting its behavior. The radical form also helps in comparing solutions and performing further algebraic manipulations without introducing approximation errors. It's a testament to the power of algebra to represent complex relationships and values in a clear and structured manner.

In conclusion, finding the zeros of the function $f(x)=x^2+8 x+4$ led us to apply the quadratic formula, yielding x = rac{-8 rac{plusmnrac{right paren rac{48}}{2}}{}. Through careful simplification of the radical rac{right paren rac{48}}{} to 4rac{right paren rac{3}}{} and subsequent division by 2, we arrived at the simplest radical form: x=-4 rac{plusmnrac{right paren rac{2rac{right paren rac{3}}{}}{}}{}. This process highlights the importance of understanding algebraic manipulation and the conventions of mathematical notation.

For further exploration into quadratic equations and functions, you can visit resources like Khan Academy for comprehensive lessons and practice problems.