Finding Roots Of X^2 - 11x + 13 A Step-by-Step Guide
In this article, we will delve into the process of finding the roots, or solutions, of the quadratic polynomial equation x_² - 11_x + 13 = 0. This is a fundamental concept in algebra, with applications spanning various fields of mathematics, science, and engineering. Understanding how to solve quadratic equations is crucial for tackling more complex problems. We will explore the quadratic formula, a powerful tool that allows us to determine the roots of any quadratic equation in the form _ax_² + bx + c = 0. The roots represent the values of x that make the equation true, and they often correspond to the points where the parabola represented by the quadratic equation intersects the x-axis. Let's begin by revisiting the quadratic formula and how it's derived. We will then apply this formula to our specific polynomial, x² - 11x + 13, and identify the two values of x that satisfy the equation. Furthermore, we'll discuss the nature of these roots, whether they are real or complex, and what that implies about the graph of the quadratic function. By the end of this article, you will have a solid understanding of how to solve quadratic equations and interpret their roots, along with a clear understanding of the solution to the given problem.
The quadratic formula is a fundamental tool in algebra for solving quadratic equations, which are equations of the form _ax_² + bx + c = 0, where a, b, and c are constants and a ≠ 0. This formula provides a direct method for finding the roots, or solutions, of the equation, which are the values of x that satisfy the equation. The quadratic formula is derived by completing the square on the general form of the quadratic equation. This involves manipulating the equation algebraically to create a perfect square trinomial on one side, which can then be easily factored. The process of completing the square involves several steps, including dividing the equation by the leading coefficient a, adding and subtracting a constant term to create the perfect square, and then solving for x. The result of this process is the quadratic formula itself, which expresses the roots of the equation in terms of the coefficients a, b, and c. The formula is given by:
x = (-b ± √(b_² - 4_ac)) / (2_a_)
In this formula: x represents the roots of the quadratic equation. The plus-minus symbol (±) indicates that there are generally two roots, one obtained by adding the square root term and the other by subtracting it. The term inside the square root, b_² - 4_ac, is known as the discriminant. The discriminant plays a crucial role in determining the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has one real root (a repeated root). And if it is negative, the equation has two complex roots. Understanding the discriminant allows us to quickly determine the type of solutions we can expect before even applying the full quadratic formula. The quadratic formula is an invaluable tool because it provides a guaranteed way to find the roots of any quadratic equation, regardless of whether the equation can be easily factored or not. It is a cornerstone of algebra and is widely used in various mathematical and scientific applications.
Now, let's apply the quadratic formula to the specific polynomial equation x_² - 11_x + 13 = 0. To do this, we first need to identify the coefficients a, b, and c in the general form of the quadratic equation, _ax_² + bx + c = 0. In our case, we have:
- a = 1 (the coefficient of _x_²)
- b = -11 (the coefficient of x)
- c = 13 (the constant term)
Now that we have identified these coefficients, we can substitute them into the quadratic formula:
x = (-b ± √(b_² - 4_ac)) / (2_a_)
Substituting our values, we get:
x = (-(-11) ± √((-11)² - 4 * 1 * 13)) / (2 * 1)
Now, we simplify the expression step by step. First, we simplify the negative of -11, which is 11. Then, we calculate the value inside the square root:
(-11)² = 121 4 * 1 * 13 = 52 121 - 52 = 69
So, the expression under the square root simplifies to 69. Now, we can rewrite the equation as:
x = (11 ± √69) / 2
This gives us two possible solutions for x: one where we add the square root of 69 and one where we subtract it. These two solutions are:
_x_₁ = (11 + √69) / 2 _x_₂ = (11 - √69) / 2
These are the two roots of the quadratic equation x_² - 11_x + 13 = 0. These roots are real numbers because the discriminant (69) is positive. They represent the points where the parabola described by the equation intersects the x-axis. By applying the quadratic formula and carefully simplifying the resulting expression, we have successfully found the two values of x that satisfy the given equation. These solutions are exact and can be used for further analysis or calculations involving this quadratic equation.
Having calculated the two roots of the quadratic equation x_² - 11_x + 13 = 0, which are:
_x_₁ = (11 + √69) / 2 _x_₂ = (11 - √69) / 2
We now need to match these solutions with the options provided in the question. The options are:
A. x = 2.5 B. x = (11 - √(-109)) / 4 C. x = (11 - √69) / 2 D. x = (11 + √(-109)) / 4 E. x = 3 F. x = (11 + √69) / 2
By comparing our calculated roots with the options, we can see that:
- _x_₂ = (11 - √69) / 2 matches with option C.
- _x_₁ = (11 + √69) / 2 matches with option F.
Therefore, the two values of x that are roots of the polynomial equation x_² - 11_x + 13 = 0 are given by options C and F. The other options can be ruled out as follows:
- Options A (x = 2.5) and E (x = 3) are numerical values that do not match our calculated roots. We could substitute these values back into the original equation to confirm that they are not solutions, but this is not necessary as they do not match our calculated expressions.
- Options B (x = (11 - √(-109)) / 4) and D (x = (11 + √(-109)) / 4) involve the square root of a negative number, which means they are complex numbers. Our calculated roots are real numbers, so these options cannot be correct. The presence of the square root of a negative number indicates that the discriminant (b_² - 4_ac) would be negative, which is not the case for our equation, where the discriminant is 69.
By carefully comparing our solutions with the provided options, we have confidently identified the correct answers as C and F. This process demonstrates the importance of accurately applying the quadratic formula and then comparing the results with the given choices to arrive at the correct solution.
As we found the roots of the quadratic equation x_² - 11_x + 13 = 0 to be (11 + √69) / 2 and (11 - √69) / 2, it is important to understand the nature of these roots and what they imply about the quadratic function represented by the equation. The nature of the roots of a quadratic equation is determined by the discriminant, which is the part of the quadratic formula under the square root: b_² - 4_ac. In our case, the discriminant is (-11)² - 4 * 1 * 13 = 121 - 52 = 69. Since the discriminant is positive (69 > 0), the equation has two distinct real roots. This means that the parabola represented by the quadratic function y = x_² - 11_x + 13 intersects the x-axis at two different points. These points of intersection are precisely the roots we calculated. If the discriminant had been zero, the equation would have had one real root (a repeated root), and the parabola would have touched the x-axis at only one point (the vertex of the parabola). If the discriminant had been negative, the equation would have had two complex roots, and the parabola would not have intersected the x-axis at all. The roots being real numbers means they can be located on the number line. The values (11 + √69) / 2 and (11 - √69) / 2 are approximately 9.65 and 1.35, respectively. These values tell us exactly where the parabola crosses the x-axis. Understanding the nature of the roots helps us visualize the graph of the quadratic function. A positive leading coefficient (a = 1 in our case) means the parabola opens upwards. Knowing that there are two distinct real roots, we can sketch a parabola that opens upwards and intersects the x-axis at two points, approximately at x = 1.35 and x = 9.65. The vertex of the parabola will lie somewhere between these two roots. In summary, the positive discriminant tells us that the quadratic equation has two distinct real roots, which correspond to the x-intercepts of the parabola. This understanding allows us to not only solve the equation but also visualize and interpret the behavior of the corresponding quadratic function.
In conclusion, we have successfully found the two values of x that are roots of the polynomial equation x_² - 11_x + 13 = 0 by applying the quadratic formula. We identified the coefficients a, b, and c, substituted them into the formula, and simplified the expression to obtain the roots (11 + √69) / 2 and (11 - √69) / 2. These roots corresponded to options C and F in the given choices. We also discussed the significance of the discriminant in determining the nature of the roots. The positive discriminant (69) indicated that the equation has two distinct real roots, which means the parabola represented by the quadratic function intersects the x-axis at two points. This understanding allows us to not only solve the equation but also visualize and interpret the behavior of the corresponding quadratic function. The quadratic formula is a powerful tool in algebra, providing a direct method for solving any quadratic equation. By mastering this formula and understanding the implications of the discriminant, we can effectively tackle a wide range of problems involving quadratic equations. The ability to find the roots of a polynomial is a fundamental skill in mathematics and has applications in various fields, including physics, engineering, and economics. This exercise has reinforced our understanding of quadratic equations and the techniques used to solve them, further solidifying our mathematical foundation. Understanding the roots of polynomials is essential for understanding more advanced mathematical concepts. The skills and knowledge gained from this exercise will be valuable in tackling more complex mathematical problems in the future.
