Solving The Quadratic Equation X(2x + 3) = X^2 + 1 A Step-by-Step Guide

In this article, we will delve into the process of solving a quadratic equation. Specifically, we will focus on the equation x(2x + 3) = x² + 1. Quadratic equations, characterized by the presence of a squared term, are fundamental in various fields of mathematics, physics, and engineering. Understanding how to solve them is a crucial skill. This detailed exploration will cover the steps required to transform the equation into its standard form, identify the coefficients, and apply the quadratic formula to find the solutions. Furthermore, we will verify these solutions by substituting them back into the original equation, ensuring accuracy and a comprehensive understanding of the solution process. Mastering the techniques to solve such equations is not only essential for academic success but also for practical applications in various scientific and technical domains.

To effectively solve the equation x(2x + 3) = x² + 1, it is essential to first understand the nature of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (in this case, x) is 2. The standard form of a quadratic equation is given by ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The coefficients a, b, and c play a crucial role in determining the nature and values of the solutions to the equation. The term ax² is known as the quadratic term, bx is the linear term, and c is the constant term. Recognizing and manipulating quadratic equations into this standard form is the first step in solving them. The solutions, also known as roots, represent the values of x that satisfy the equation. These roots can be real or complex numbers, depending on the discriminant (b² - 4ac) of the equation. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real root (or two equal real roots), and a negative discriminant indicates two complex roots. Therefore, a thorough understanding of the standard form and the nature of the coefficients is crucial for accurately solving quadratic equations.

The first crucial step in solving the equation x(2x + 3) = x² + 1 is to transform it into the standard quadratic form ax² + bx + c = 0. This process involves several algebraic manipulations to rearrange the terms appropriately. Initially, the equation is given as x(2x + 3) = x² + 1. To begin, we need to expand the left side of the equation by distributing x across the terms inside the parenthesis. This gives us 2x² + 3x = x² + 1. Next, our goal is to bring all the terms to one side of the equation, leaving zero on the other side. To achieve this, we subtract x² from both sides of the equation, resulting in 2x² - x² + 3x = 1. Simplifying this further, we get x² + 3x = 1. Now, to complete the transformation to the standard form, we subtract 1 from both sides of the equation, which yields x² + 3x - 1 = 0. This equation is now in the standard quadratic form, ax² + bx + c = 0, where a = 1, b = 3, and c = -1. Transforming the equation into this standard form is a critical step because it allows us to easily identify the coefficients a, b, and c, which are necessary for applying methods such as the quadratic formula to find the solutions.

After transforming the equation x(2x + 3) = x² + 1 into its standard form, x² + 3x - 1 = 0, the next critical step is to accurately identify the coefficients a, b, and c. These coefficients are the numerical values associated with the terms in the quadratic equation and are crucial for applying the quadratic formula or other methods to find the solutions. In the standard form equation ax² + bx + c = 0, the coefficient a is the number multiplying the x² term, b is the number multiplying the x term, and c is the constant term. In our specific equation, x² + 3x - 1 = 0, we can easily identify these coefficients by comparing it to the standard form. 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 the x term, which is 3. Lastly, the coefficient c is the constant term, which is -1. Thus, we have a = 1, b = 3, and c = -1. Correctly identifying these coefficients is paramount because they are directly used in the quadratic formula, and any mistake in their identification will lead to incorrect solutions. This careful identification ensures that the subsequent steps in solving the equation will be accurate, leading to the correct roots of the quadratic equation.

With the coefficients a, b, and c identified as a = 1, b = 3, and c = -1, the next pivotal step in solving the quadratic equation x(2x + 3) = x² + 1 is to apply the quadratic formula. The quadratic formula is a universal method for finding the solutions (or roots) of any quadratic equation in the standard form ax² + bx + c = 0. The formula is given by: x = (-b ± √(b² - 4ac)) / (2a). This formula provides two possible solutions for x, one using the plus sign (+) and the other using the minus sign (±). To apply the quadratic formula, we substitute the identified values of a, b, and c into the formula. In our case, substituting a = 1, b = 3, and c = -1 gives us: x = (-3 ± √(3² - 4(1)(-1))) / (2(1)). We then simplify the expression step by step. First, we calculate the term inside the square root: 3² is 9, and 4(1)(-1) is -4. So, the expression inside the square root becomes 9 - (-4), which simplifies to 9 + 4 = 13. The formula now looks like: x = (-3 ± √13) / 2. This results in two potential solutions for x: x = (-3 + √13) / 2 and x = (-3 - √13) / 2. These are the exact solutions to the quadratic equation. The quadratic formula provides a systematic approach to finding these solutions, ensuring accuracy and completeness in the solution process. Understanding and correctly applying this formula is crucial for solving a wide range of quadratic equations.

Having applied the quadratic formula to the equation x(2x + 3) = x² + 1, we arrived at the solutions x = (-3 + √13) / 2 and x = (-3 - √13) / 2. Now, we need to calculate the approximate numerical values of these solutions. The square root of 13 (√13) is an irrational number, meaning it cannot be expressed as a simple fraction and has a non-repeating, non-terminating decimal representation. The approximate value of √13 is 3.60555. To find the numerical solutions, we will substitute this approximate value into our expressions for x. For the first solution, x = (-3 + √13) / 2, we substitute √13 with 3.60555, giving us x = (-3 + 3.60555) / 2. Simplifying the numerator, we get 0.60555 / 2, which results in x ≈ 0.30278. For the second solution, x = (-3 - √13) / 2, we again substitute √13 with 3.60555, resulting in x = (-3 - 3.60555) / 2. The numerator simplifies to -6.60555 / 2, which gives us x ≈ -3.30278. Therefore, the approximate solutions to the quadratic equation are x ≈ 0.30278 and x ≈ -3.30278. These values are the points where the quadratic function intersects the x-axis, providing the roots of the equation. Calculating these numerical solutions helps in visualizing the equation's behavior and understanding its practical implications.

After calculating the solutions for the equation x(2x + 3) = x² + 1, the final essential step is to verify these solutions. Verification involves substituting the calculated values of x back into the original equation to ensure they satisfy it. This process confirms the accuracy of our calculations and the correctness of the solutions. We found two solutions: x ≈ 0.30278 and x ≈ -3.30278. Let's start by verifying the first solution, x ≈ 0.30278. Substituting this value into the original equation, we get: 0.30278(2(0.30278) + 3) = (0.30278)² + 1. First, we simplify the left side of the equation: 2(0.30278) is approximately 0.60556, so the equation becomes 0.30278(0.60556 + 3). Adding the terms inside the parenthesis gives us 0.30278(3.60556), which is approximately 1.09169. Next, we simplify the right side of the equation: (0.30278)² is approximately 0.09167, so the equation becomes 0.09167 + 1, which simplifies to 1.09167. Comparing both sides, we see that 1.09169 is very close to 1.09167, indicating that the solution x ≈ 0.30278 is valid. Now, let's verify the second solution, x ≈ -3.30278. Substituting this value into the original equation, we get: -3.30278(2(-3.30278) + 3) = (-3.30278)² + 1. First, we simplify the left side of the equation: 2(-3.30278) is approximately -6.60556, so the equation becomes -3.30278(-6.60556 + 3). Adding the terms inside the parenthesis gives us -3.30278(-3.60556), which is approximately 11.9078. Next, we simplify the right side of the equation: (-3.30278)² is approximately 10.9084, so the equation becomes 10.9084 + 1, which simplifies to 11.9084. Comparing both sides, we see that 11.9078 is very close to 11.9084, indicating that the solution x ≈ -3.30278 is also valid. The slight differences in the results are due to rounding during the calculations. Overall, the verification process confirms that both solutions are correct, reinforcing the accuracy of our method and the validity of the solutions found.

In conclusion, we have successfully solved the quadratic equation x(2x + 3) = x² + 1 through a step-by-step process, demonstrating a comprehensive approach to handling such equations. Initially, the equation was transformed into its standard form, x² + 3x - 1 = 0, which is a crucial step for applying standard solution methods. Following this, we identified the coefficients a, b, and c as 1, 3, and -1, respectively. These coefficients were then used in the quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), to find the exact solutions. The application of the quadratic formula led us to the solutions x = (-3 + √13) / 2 and x = (-3 - √13) / 2. We then calculated the approximate numerical values of these solutions as x ≈ 0.30278 and x ≈ -3.30278. To ensure the accuracy of our results, we verified these solutions by substituting them back into the original equation, confirming that they indeed satisfy the equation. This verification process is a critical step in problem-solving, as it helps to catch any potential errors in the calculations. The solutions obtained represent the roots of the quadratic equation, which are the points where the parabola defined by the equation intersects the x-axis. This entire process underscores the importance of understanding the properties of quadratic equations, the correct application of the quadratic formula, and the necessity of verifying solutions. Mastering these skills is essential for further studies in mathematics and its applications in various fields such as physics, engineering, and computer science. By meticulously following these steps, one can confidently solve a wide range of quadratic equations, gaining a deeper understanding of this fundamental concept in algebra.