Finding The Zeros Of A Quadratic Function

When we talk about the zeros of a function, we're essentially asking: "At what x-values does the function's output (y-value) equal zero?" For a quadratic function, specifically one in the form of $f(x) = ax^2 + bx + c$, finding these zeros means solving the equation $ax^2 + bx + c = 0$. This process is fundamental in understanding the behavior of parabolas, which are the graphical representations of quadratic functions. The zeros tell us where the parabola intersects the x-axis. In our case, we're looking at the specific quadratic function $f(x) = x^2 - x - 20$. Our goal is to find the values of $x$ that make $f(x)$ equal to zero. There are several ways to tackle this. One common and powerful method is factoring the quadratic expression. Factoring involves rewriting the quadratic as a product of two linear expressions. If we can do this, say $f(x) = (x - r_1)(x - r_2)$, then setting $f(x) = 0$ means $(x - r_1)(x - r_2) = 0$. By the zero product property, this equation holds true if either $x - r_1 = 0$ (which means $x = r_1$) or $x - r_2 = 0$ (which means $x = r_2$). These values, $r_1$ and $r_2$, are the zeros of the function. Another widely applicable method is using the quadratic formula, which provides a direct solution for $x$ for any quadratic equation of the form $ax^2 + bx + c = 0$. The formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula is incredibly useful, especially when factoring seems difficult or impossible. For our function $f(x) = x^2 - x - 20$, we have $a=1$, $b=-1$, and $c=-20$. Plugging these values into the quadratic formula would also yield the solutions. Let's explore the factoring method first, as it often provides a more intuitive understanding when it works. We need to find two numbers that multiply to $c$ (which is -20) and add up to $b$ (which is -1). Let's list the pairs of factors of -20: (1, -20), (-1, 20), (2, -10), (-2, 10), (4, -5), and (-4, 5). Now, let's check the sum of each pair: 1 + (-20) = -19; -1 + 20 = 19; 2 + (-10) = -8; -2 + 10 = 8; 4 + (-5) = -1; -4 + 5 = 1. Bingo! The pair (4, -5) adds up to -1. This means we can factor our quadratic expression $x^2 - x - 20$ into $(x+4)(x-5)$. Therefore, to find the zeros, we set $(x+4)(x-5) = 0$. Using the zero product property, we have two possibilities: $x+4=0$ or $x-5=0$. Solving these simple linear equations, we get $x=-4$ and $x=5$. These are the two values of $x$ for which $f(x)$ equals zero. These are the points where the parabola representing $f(x)=x^2-x-20$ crosses the x-axis.

Let's delve deeper into why understanding the zeros of a quadratic function, like $f(x) = x^2 - x - 20$, is so crucial in mathematics and various applications. The zeros of a function are also known as its roots or x-intercepts. They represent the points where the graph of the function crosses or touches the x-axis. For quadratic functions, these zeros provide invaluable information about the parabola's position and shape. For instance, if a quadratic function has two distinct real zeros, its parabola intersects the x-axis at two different points. If it has one real zero (a repeated root), the parabola touches the x-axis at exactly one point, its vertex. If it has no real zeros, the parabola either lies entirely above or entirely below the x-axis, never intersecting it. In our specific case, with $f(x) = x^2 - x - 20$, we found the zeros to be $x=-4$ and $x=5$. This tells us that the parabola representing this function intersects the x-axis at the points $(-4, 0)$ and $(5, 0)$. This information is vital for sketching the graph accurately. Furthermore, the zeros help in analyzing the sign of the function. The quadratic expression $x^2 - x - 20$ will be positive for x-values outside the interval $(-4, 5)$ and negative for x-values within this interval (since the coefficient of $x^2$ is positive, meaning the parabola opens upwards). This understanding of intervals where the function is positive or negative is essential in solving inequalities and in optimization problems. The process of finding these zeros can be approached through different algebraic techniques. While factoring is often the most straightforward when applicable, it's not always possible to easily factor a quadratic. This is where the quadratic formula shines. For $f(x) = x^2 - x - 20$, $a=1$, $b=-1$, and $c=-20$. Using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, we substitute the values: $x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-20)}}{2(1)}$. This simplifies to $x = \frac{1 \pm \sqrt{1 + 80}}{2}$, which is $x = \frac{1 \pm \sqrt{81}}{2}$. Taking the square root of 81, we get 9. So, $x = \frac{1 \pm 9}{2}$. This gives us two solutions: $x_1 = \frac{1 + 9}{2} = \frac{10}{2} = 5$ and $x_2 = \frac{1 - 9}{2} = \frac{-8}{2} = -4$. These results perfectly match the ones we obtained through factoring, reinforcing their correctness. Another method, though less common for finding exact zeros, is completing the square. This involves manipulating the equation $x^2 - x - 20 = 0$ into the form $(x-h)^2 = k$. Starting with $x^2 - x = 20$, we take half of the coefficient of $x$ (which is -1), square it ($\left(-\frac{1}{2}\right)^2 = \frac{1}{4}$), and add it to both sides: $x^2 - x + \frac{1}{4} = 20 + \frac{1}{4}$. The left side is now a perfect square: $\left(x - \frac{1}{2}\right)^2 = \frac{81}{4}$. Taking the square root of both sides, $x - \frac{1}{2} = \pm \sqrt{\frac{81}{4}} = \pm \frac{9}{2}$. Solving for $x$: $x = \frac{1}{2} \pm \frac{9}{2}$. This again yields $x_1 = \frac{1}{2} + \frac{9}{2} = \frac{10}{2} = 5$ and $x_2 = \frac{1}{2} - \frac{9}{2} = \frac{-8}{2} = -4$. All three methods converge on the same answer, which is a testament to the robustness of mathematical principles. The ability to find the zeros of quadratic functions is a cornerstone for understanding more complex mathematical concepts and is widely used in fields such as physics (e.g., projectile motion), engineering, economics, and computer graphics.

Let's analyze the provided options in relation to our findings for the zeros of $f(x) = x^2 - x - 20$. We've established through factoring, the quadratic formula, and completing the square that the zeros of this function are $x = -4$ and $x = 5$. Now, we'll systematically examine each option to see which one correctly identifies these zeros. Remember, the zeros are the specific x-values where the function's output is zero. We are looking for a pair of numbers that, when substituted into $f(x)$, result in $f(x)=0$. The question is designed to test your understanding of how to solve for these roots and to recognize the correct pair among common distractors. It's important to not just find the zeros but also to correctly match them to the options provided. The options presented are:

A. $x=-2$ and $x=10$ B. $x=-4$ and $x=5$ C. $x=-5$ and $x=4$ D. $x=-10$ and $x=2$

Let's test each option. For option A, if $x=-2$, $f(-2) = (-2)^2 - (-2) - 20 = 4 + 2 - 20 = 6 - 20 = -14 \neq 0$. If $x=10$, $f(10) = (10)^2 - (10) - 20 = 100 - 10 - 20 = 90 - 20 = 70 \neq 0$. So, option A is incorrect.

For option B, if $x=-4$, $f(-4) = (-4)^2 - (-4) - 20 = 16 + 4 - 20 = 20 - 20 = 0$. This is a correct zero. If $x=5$, $f(5) = (5)^2 - (5) - 20 = 25 - 5 - 20 = 20 - 20 = 0$. This is also a correct zero. Since both values work, option B correctly lists the zeros of the function.

For option C, if $x=-5$, $f(-5) = (-5)^2 - (-5) - 20 = 25 + 5 - 20 = 30 - 20 = 10 \neq 0$. If $x=4$, $f(4) = (4)^2 - (4) - 20 = 16 - 4 - 20 = 12 - 20 = -8 \neq 0$. So, option C is incorrect.

For option D, if $x=-10$, $f(-10) = (-10)^2 - (-10) - 20 = 100 + 10 - 20 = 110 - 20 = 90 \neq 0$. If $x=2$, $f(2) = (2)^2 - (2) - 20 = 4 - 2 - 20 = 2 - 20 = -18 \neq 0$. So, option D is incorrect.

As we can see, only option B provides the correct pair of zeros for the function $f(x) = x^2 - x - 20$. The distractors (options A, C, and D) often arise from common errors in factoring or applying the quadratic formula, such as incorrect signs or arithmetic mistakes. For instance, option C has the numbers 4 and -5, which are the factors we found, but they are presented as the zeros instead of the roots derived from those factors. Option A and D present pairs that might come from miscalculations in the quadratic formula or incorrect factoring attempts. It's a good practice to always double-check your answers by substituting them back into the original function, as we've done here, to ensure accuracy. This methodical approach guarantees that you arrive at the correct solution and understand the underlying mathematical principles. The ability to find zeros is a foundational skill in algebra, opening doors to understanding more advanced topics in calculus and applied mathematics.

Conclusion

In conclusion, finding the zeros of a quadratic function like $f(x) = x^2 - x - 20$ is a fundamental skill in mathematics. These zeros, also known as roots or x-intercepts, represent the points where the function's graph intersects the x-axis. We explored various methods to find these zeros, including factoring, the quadratic formula, and completing the square. For $f(x) = x^2 - x - 20$, we discovered that the expression can be factored into $(x+4)(x-5)$. Setting this equal to zero, $(x+4)(x-5) = 0$, and applying the zero product property, we found the zeros to be $x = -4$ and $x = 5$. We then systematically evaluated the given options, confirming that only option B correctly lists these zeros. This detailed analysis ensures not only the correct answer but also a deeper understanding of the concepts involved. Mastering the techniques for finding quadratic zeros is essential for solving a wide range of mathematical problems and for applications in various scientific and engineering fields.

For further exploration into quadratic functions and their properties, you can consult resources like ** **Khan Academy's Algebra Section. They offer comprehensive explanations, practice exercises, and video tutorials that can enhance your understanding of this topic and related mathematical concepts.