Solutions Of The Quadratic Equation X² = -11x + 4

In this article, we will explore the solutions to the quadratic equation x² = -11x + 4. Quadratic equations, which take the general form ax² + bx + c = 0, are fundamental in algebra and have wide-ranging applications in various fields such as physics, engineering, and economics. Finding the solutions, also known as roots or zeros, of a quadratic equation involves determining the values of x that satisfy the equation. We will employ the quadratic formula, a powerful tool for solving these equations, and meticulously analyze each step to ensure clarity and understanding. The quadratic formula allows us to find the roots of any quadratic equation, regardless of whether it can be easily factored. Understanding how to apply this formula is crucial for solving a multitude of mathematical problems. We will begin by rearranging the given equation into the standard quadratic form, which will make it easier to identify the coefficients a, b, and c. These coefficients are essential components of the quadratic formula, and correctly identifying them is the first step towards finding the solutions. Next, we will substitute these coefficients into the quadratic formula and simplify the expression step by step. This process will involve dealing with square roots and fractions, so we will pay close attention to detail to avoid any errors. Finally, we will interpret the results to determine the two possible solutions for x. These solutions represent the points where the parabola defined by the quadratic equation intersects the x-axis. Through this detailed exploration, we aim to provide a comprehensive understanding of how to solve quadratic equations using the quadratic formula, thereby enhancing your problem-solving skills in algebra and beyond.

Transforming the Equation

The initial step in solving the quadratic equation is to transform it into the standard form, which is ax² + bx + c = 0. Our equation is given as x² = -11x + 4. To bring it into the standard form, we need to move all terms to one side of the equation, leaving zero on the other side. This involves adding 11x to both sides and subtracting 4 from both sides. By performing these operations, we will have all the terms on the left side, making it easy to identify the coefficients a, b, and c. The transformation process is a critical step because it sets the stage for applying the quadratic formula correctly. Ensuring that the equation is in the standard form is essential for accurately identifying the coefficients, which are the building blocks for the subsequent steps. Let's begin the transformation by adding 11x to both sides of the equation. This eliminates the -11x term from the right side, moving it to the left side of the equation. Next, we subtract 4 from both sides, which moves the constant term to the left side and leaves zero on the right side. The resulting equation will be in the form ax² + bx + c = 0, where a, b, and c are constants. This standard form allows us to directly apply the quadratic formula, which is a general solution for quadratic equations. By meticulously following this transformation process, we ensure that the equation is in the correct format for solving, which is a crucial step in obtaining accurate solutions. The standard form not only simplifies the application of the quadratic formula but also aids in understanding the structure and properties of quadratic equations. This structured approach is fundamental to solving various types of mathematical problems, making it an essential skill for any student of algebra.

The transformed equation is:

x² + 11x - 4 = 0

Identifying Coefficients

Now that we have the equation in the standard form, x² + 11x - 4 = 0, the next step is to identify the coefficients a, b, and c. These coefficients are the numerical values that multiply the terms x², x, and the constant term, respectively. In our equation, the coefficient a is the number multiplying x², which is 1 (since x² is the same as 1x²). The coefficient b is the number multiplying x, which is 11. The coefficient c is the constant term, which is -4. Correctly identifying these coefficients is crucial because they are directly used in the quadratic formula. Any error in identifying the coefficients will lead to incorrect solutions. Therefore, it is important to pay close attention to the signs and numerical values of each term. The coefficients a, b, and c play different roles in determining the shape and position of the parabola represented by the quadratic equation. The coefficient a determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0), and it also affects the width of the parabola. The coefficient b influences the position of the parabola's axis of symmetry, and the coefficient c determines the y-intercept of the parabola. Understanding the significance of these coefficients can provide valuable insights into the behavior of quadratic functions. By accurately identifying a = 1, b = 11, and c = -4, we are well-prepared to apply the quadratic formula and find the solutions to the equation. This step is a foundational element in solving quadratic equations, and mastering it ensures a solid understanding of the process.

• a = 1
• b = 11
• c = -4

Applying the Quadratic Formula

The quadratic formula is a powerful tool for finding the solutions to any quadratic equation in the form ax² + bx + c = 0. The formula is given by: x = (-b ± √(b² - 4ac)) / (2a). This formula provides two possible solutions for x, which correspond to the points where the parabola defined by the quadratic equation intersects the x-axis. To apply the quadratic formula, we substitute the values of the coefficients a, b, and c that we identified in the previous step. In our case, a = 1, b = 11, and c = -4. Substituting these values into the quadratic formula, we get: x = (-11 ± √(11² - 4(1)(-4))) / (2(1)). The next step is to simplify the expression under the square root and the denominator. We first calculate b², which is 11² = 121. Then we calculate 4ac, which is 4(1)(-4) = -16. Subtracting 4ac from b² gives us 121 - (-16) = 121 + 16 = 137. The denominator is 2a, which is 2(1) = 2. So, the quadratic formula now looks like this: x = (-11 ± √137) / 2. This expression gives us two solutions, one with the plus sign and one with the minus sign. The square root of 137 cannot be simplified further, as 137 is a prime number. Therefore, the solutions are expressed in terms of the square root. The quadratic formula is a fundamental concept in algebra, and understanding how to apply it is essential for solving a wide range of problems. It provides a systematic approach to finding the roots of quadratic equations, regardless of whether they can be easily factored. By carefully substituting the coefficients and simplifying the expression, we can accurately determine the solutions.

x = (-b ± √(b² - 4ac)) / (2a)

Substituting the values a = 1, b = 11, and c = -4:

x = (-11 ± √(11² - 4(1)(-4))) / (2(1))

Simplifying the Expression

After substituting the coefficients into the quadratic formula, we now need to simplify the expression to find the solutions for x. Our expression is x = (-11 ± √(11² - 4(1)(-4))) / (2(1)). The first step in simplifying is to evaluate the terms inside the square root and the denominator. Inside the square root, we have 11² - 4(1)(-4). We first calculate 11², which is 121. Then we calculate 4(1)(-4), which is -16. So, the expression inside the square root becomes 121 - (-16), which simplifies to 121 + 16 = 137. The denominator is 2(1), which simplifies to 2. Now, our expression looks like this: x = (-11 ± √137) / 2. The square root of 137 cannot be simplified further because 137 is a prime number. Therefore, we leave it as √137. The expression now gives us two solutions: x = (-11 + √137) / 2 and x = (-11 - √137) / 2. These are the exact solutions to the quadratic equation. Simplifying the expression is a critical step in solving quadratic equations because it allows us to express the solutions in their simplest form. It involves carefully following the order of operations and paying attention to details such as signs and square roots. The ability to simplify complex expressions is a fundamental skill in algebra and is essential for solving a wide range of mathematical problems. By systematically breaking down the expression and evaluating each term, we can arrive at the final solutions. In this case, the solutions are expressed in terms of the square root of 137, which provides an accurate representation of the values of x that satisfy the equation. This process demonstrates the power of the quadratic formula in finding solutions to quadratic equations, even when the roots are not rational numbers.

x = (-11 ± √(121 + 16)) / 2 x = (-11 ± √137) / 2

Final Solutions

After simplifying the expression, we have two solutions for x: x = (-11 + √137) / 2 and x = (-11 - √137) / 2. These solutions represent the roots of the quadratic equation x² + 11x - 4 = 0. The first solution, x = (-11 + √137) / 2, is obtained by using the plus sign in the ± symbol. The second solution, x = (-11 - √137) / 2, is obtained by using the minus sign. These solutions are exact and provide the values of x that make the equation true. They are expressed in terms of the square root of 137, which is an irrational number. This means that the solutions cannot be expressed as simple fractions or terminating decimals. The presence of the square root indicates that the roots are not rational, and the decimal representation would be non-repeating and non-terminating. The solutions can be approximated using a calculator to obtain decimal values, but the exact solutions are best represented in this form. Understanding that quadratic equations can have two distinct solutions, one solution (a repeated root), or no real solutions (complex roots) is crucial in algebra. In this case, we have two distinct real solutions. These solutions are the x-intercepts of the parabola represented by the equation y = x² + 11x - 4. The x-intercepts are the points where the parabola crosses the x-axis, and they provide valuable information about the behavior of the quadratic function. By finding these solutions, we have completely solved the quadratic equation and gained a deeper understanding of its properties. This process reinforces the importance of the quadratic formula as a fundamental tool in algebra, allowing us to solve a wide range of problems and applications.

Therefore, the solutions are:

x₁ = (-11 + √137) / 2 x₂ = (-11 - √137) / 2

Conclusion

In this article, we have thoroughly explored the process of finding the solutions to the quadratic equation x² = -11x + 4. We began by transforming the equation into the standard form, ax² + bx + c = 0, which is essential for applying the quadratic formula. We identified the coefficients a = 1, b = 11, and c = -4, and then substituted these values into the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). By carefully simplifying the expression, we arrived at the two solutions: x₁ = (-11 + √137) / 2 and x₂ = (-11 - √137) / 2. These solutions represent the values of x that satisfy the original equation and are expressed in terms of the square root of 137. The quadratic formula is a fundamental tool in algebra, providing a systematic way to solve quadratic equations, regardless of whether they can be easily factored. It is a powerful method for finding the roots of quadratic equations and is applicable in a wide range of mathematical and real-world contexts. Understanding the steps involved in applying the quadratic formula, from transforming the equation to simplifying the expression, is crucial for mastering algebra. This process not only helps in finding the solutions but also provides insights into the properties of quadratic equations and their graphical representations. The ability to solve quadratic equations is a cornerstone of mathematical literacy and is essential for further studies in mathematics, science, and engineering. By following the steps outlined in this article, you can confidently tackle quadratic equations and enhance your problem-solving skills. The quadratic formula is a versatile and reliable tool that will serve you well in your mathematical journey, allowing you to solve a variety of problems and deepen your understanding of algebraic concepts.