Solving Quadratic Equations Using The Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠0. This article will guide you through the process of using the quadratic formula to solve a specific quadratic equation, providing a clear and detailed explanation of each step.
Understanding the Quadratic Formula
The quadratic formula is derived from the method of completing the square and provides a general solution for any quadratic equation. The formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
Where:
- x represents the solutions (also called roots) of the quadratic equation.
- a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.
- The ± symbol indicates that there are two possible solutions: one with addition and one with subtraction.
- The expression b² - 4ac is called the discriminant, which determines the nature of the roots (real or complex).
Before diving into the example, let's emphasize the importance of understanding the quadratic formula. It's not just about memorizing the formula, but also about grasping its underlying principles and how it's derived. This understanding will allow you to apply the formula effectively in various contexts and solve a wide range of quadratic equations. The quadratic formula ensures we find all possible solutions, even those that are not immediately obvious through factoring or other methods. It's a cornerstone of algebra and essential for further studies in mathematics and related fields. By mastering the quadratic formula, you're equipping yourself with a valuable problem-solving skill that will serve you well in your academic and professional journey. Therefore, take the time to truly understand the formula, its components, and its applications. With practice and perseverance, you'll become proficient in using it to solve any quadratic equation you encounter.
Problem Statement: Solving 4x² - 3x + 9 = 2x + 1
Our goal is to find the values of x that satisfy the quadratic equation 4x² - 3x + 9 = 2x + 1. To do this, we'll follow these steps:
- Rewrite the equation in the standard quadratic form (ax² + bx + c = 0).
- Identify the coefficients a, b, and c.
- Substitute the values of a, b, and c into the quadratic formula.
- Simplify the expression and find the solutions for x.
This particular problem requires careful attention to detail, especially when simplifying the expression under the square root. The presence of the constant terms and the linear term on both sides of the equation necessitates a preliminary step of rearranging the terms to bring the equation into the standard quadratic form. This rearrangement is crucial because the coefficients a, b, and c used in the quadratic formula must correspond to the standard form. Neglecting this step can lead to incorrect identification of the coefficients and, consequently, incorrect solutions. Furthermore, the simplification process may involve dealing with negative numbers and imaginary units, which requires a solid understanding of complex number arithmetic. Therefore, a methodical approach, with each step clearly laid out and meticulously executed, is essential for solving this problem accurately. This carefulness not only ensures the correct answer but also reinforces the importance of precision in mathematical problem-solving.
Step 1: Rewrite the Equation in Standard Form
To use the quadratic formula, we need to rewrite the equation in the standard form ax² + bx + c = 0. Start by subtracting 2x and 1 from both sides of the equation:
4x² - 3x + 9 - 2x - 1 = 2x + 1 - 2x - 1
Combine like terms:
4x² - 5x + 8 = 0
Now the equation is in the standard form, and we can proceed to the next step.
The importance of rewriting the equation in standard form cannot be overstated. This step is not merely a cosmetic rearrangement; it is a fundamental requirement for the correct application of the quadratic formula. The coefficients a, b, and c in the formula are defined specifically in relation to the standard form ax² + bx + c = 0. If the equation is not in this form, the values of a, b, and c will be incorrectly identified, leading to an erroneous result. The process of rewriting involves isolating all terms on one side of the equation, leaving zero on the other side. This often involves adding or subtracting terms from both sides and then combining like terms to simplify the expression. It's a critical step that sets the stage for the subsequent steps in the solution process. Therefore, before attempting to apply the quadratic formula, always ensure that the equation is in standard form to guarantee the accuracy of your solution.
Step 2: Identify the Coefficients
Now that the equation is in the standard form 4x² - 5x + 8 = 0, we can identify the coefficients:
- a = 4
- b = -5
- c = 8
These values will be used in the quadratic formula.
Accurately identifying the coefficients a, b, and c is a crucial step in applying the quadratic formula. These coefficients are the numerical values that multiply the x² term, the x term, and the constant term, respectively, in the standard form of the quadratic equation ax² + bx + c = 0. A mistake in identifying even one of these coefficients will lead to an incorrect solution. Pay close attention to the signs of the coefficients. For instance, if the equation is 4x² - 5x + 8 = 0, then a = 4, b = -5 (note the negative sign), and c = 8. The negative sign in front of the 5 is an integral part of the value of b. Similarly, if a term is missing, its coefficient is zero. For example, in the equation 2x² + 7 = 0, a = 2, b = 0 (since there is no x term), and c = 7. Therefore, carefully examine the equation in standard form and extract the coefficients with their correct signs to ensure the accurate application of the quadratic formula.
Step 3: Apply the Quadratic Formula
Substitute the values of a, b, and c into the quadratic formula:
x = (-(-5) ± √((-5)² - 4 * 4 * 8)) / (2 * 4)
This substitution is a straightforward process, but it requires careful attention to detail to avoid errors.
The act of substituting the identified coefficients into the quadratic formula is a mechanical yet critical step in the solution process. It involves replacing the variables a, b, and c in the formula with their corresponding numerical values, which we determined in the previous step. This substitution must be done meticulously, paying close attention to signs and order of operations. For example, in the formula -b, if b is a negative number, such as -5, then -b becomes -(-5), which simplifies to +5. Similarly, the expression b² means squaring the entire value of b, including its sign. So, if b is -5, then b² is (-5)² = 25. The same level of care must be applied to the term 4ac and the denominator 2a. A single mistake in substitution can propagate through the rest of the solution, leading to an incorrect answer. Therefore, double-check your substitutions to ensure accuracy before proceeding to the next step of simplifying the expression.
Step 4: Simplify the Expression
Simplify the expression step by step:
x = (5 ± √(25 - 128)) / 8
x = (5 ± √(-103)) / 8
Since the value inside the square root is negative, we will have complex solutions. Recall that √(-1) = i, where i is the imaginary unit.
x = (5 ± √103 * i) / 8
Thus, the solutions are:
x = (5 ± √103 * i) / 8
The simplification process following the substitution into the quadratic formula is a multi-stage endeavor that demands a solid understanding of arithmetic operations, including those involving square roots and imaginary numbers. The first step often involves simplifying the discriminant, which is the expression under the square root (b² - 4ac). This requires careful evaluation of the squares, multiplications, and subtractions. If the discriminant is negative, as in this case, it indicates that the solutions are complex numbers, which involve the imaginary unit i, where i² = -1. The square root of a negative number, such as √(-103), is expressed as √103 * i. The final step is to express the solutions in the standard form of complex numbers, which is a + bi, where a and b are real numbers. This may involve separating the real and imaginary parts of the solution and simplifying the expression as much as possible. Throughout this simplification process, it is crucial to maintain accuracy and pay attention to detail to arrive at the correct solutions. Each step builds upon the previous one, and a mistake at any point can invalidate the final result.
Final Answer
The values of x are:
x = (5 ± √103 * i) / 8
Therefore, the correct answer is:
C. (5 ± √103 * i) / 8
This detailed step-by-step solution demonstrates how to use the quadratic formula to solve a quadratic equation, even when the solutions are complex numbers. Remember to always rewrite the equation in standard form, identify the coefficients correctly, and simplify the expression carefully to arrive at the correct answer.
In conclusion, the quadratic formula is an indispensable tool for solving quadratic equations. Mastering its application requires a systematic approach, including rewriting the equation in standard form, accurately identifying coefficients, careful substitution, and meticulous simplification. The ability to handle complex numbers and imaginary units is also crucial when the discriminant is negative. By following these steps diligently and practicing regularly, you can confidently solve a wide range of quadratic equations and enhance your problem-solving skills in mathematics.
