Solving 2x² - 3x - 2 = 0 Using The Quadratic Formula
In the realm of algebra, quadratic equations stand as fundamental expressions with numerous applications in various fields, ranging from physics and engineering to economics and computer science. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and x represents the unknown variable. Solving quadratic equations involves finding the values of x that satisfy the equation, which are also known as the roots or solutions of the equation. Among the various methods for solving quadratic equations, the quadratic formula stands out as a powerful and versatile technique that can be applied to any quadratic equation, regardless of the nature of its roots. This formula provides a direct and systematic approach to finding the solutions, making it an indispensable tool in the arsenal of any mathematics enthusiast or problem solver.
The quadratic formula is derived from the process of completing the square, a technique used to transform a quadratic equation into a perfect square trinomial. By applying the quadratic formula, we can efficiently determine the roots of any quadratic equation, whether they are real or complex. In this comprehensive guide, we will delve into the application of the quadratic formula to solve the equation 2x² - 3x - 2 = 0, providing a step-by-step explanation that will enhance your understanding of this essential mathematical concept. We will explore the significance of the discriminant, which helps us determine the nature of the roots, and discuss the various scenarios that can arise when solving quadratic equations. By the end of this guide, you will be well-equipped to confidently tackle any quadratic equation using the quadratic formula.
The quadratic formula is a mathematical expression that provides the solutions to any quadratic equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
This formula reveals that a quadratic equation can have up to two distinct solutions, denoted by the ± symbol. These solutions are also referred to as the roots of the equation, representing the values of x that satisfy the equation. The expression within the square root, b² - 4ac, is known as the discriminant, and it plays a crucial role in determining the nature of the roots. The discriminant can be positive, zero, or negative, each case indicating different characteristics of the solutions.
When the discriminant is positive (b² - 4ac > 0), the quadratic equation has two distinct real roots. This means there are two different real numbers that satisfy the equation. When the discriminant is zero (b² - 4ac = 0), the quadratic equation has exactly one real root, which is sometimes referred to as a repeated root. In this case, the two solutions provided by the quadratic formula are identical. When the discriminant is negative (b² - 4ac < 0), the quadratic equation has two complex roots. Complex roots involve the imaginary unit i, where i² = -1, and they occur in conjugate pairs. Understanding the discriminant is essential for predicting the type of solutions you will obtain when solving a quadratic equation.
Before applying the quadratic formula, it is essential to correctly identify the coefficients a, b, and c from the given quadratic equation. The equation we aim to solve is 2x² - 3x - 2 = 0. By comparing this equation to the standard form of a quadratic equation, ax² + bx + c = 0, we can easily extract the values of the coefficients. 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 equation, 2x² - 3x - 2 = 0, the coefficient 'a' corresponds to the number multiplying x², which is 2. The coefficient 'b' corresponds to the number multiplying x, which is -3. Note that we include the negative sign, as it is crucial for accurate calculations. The constant term 'c' is the number without any x term, which is -2. Again, we include the negative sign. Thus, we have identified a = 2, b = -3, and c = -2. These values are the key ingredients for applying the quadratic formula to find the solutions of the equation.
Now that we have identified the coefficients a = 2, b = -3, and c = -2, we can proceed with applying the quadratic formula to solve the equation 2x² - 3x - 2 = 0. The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
The first step is to substitute the values of a, b, and c into the formula. Replacing a with 2, b with -3, and c with -2, we get:
x = (-(-3) ± √((-3)² - 4 * 2 * (-2))) / (2 * 2)
Next, we simplify the expression. The double negative in -(-3) becomes positive 3. Inside the square root, we have (-3)² which equals 9, and 4 * 2 * (-2) which equals -16. So, the expression becomes:
x = (3 ± √(9 - (-16))) / 4
Simplifying further, we have 9 - (-16) which equals 9 + 16 = 25. The square root of 25 is 5. Thus, the expression simplifies to:
x = (3 ± 5) / 4
This gives us two possible solutions, one with the plus sign and one with the minus sign. Let's calculate each solution separately.
For the solution with the plus sign:
x = (3 + 5) / 4 = 8 / 4 = 2
For the solution with the minus sign:
x = (3 - 5) / 4 = -2 / 4 = -1/2
Therefore, the solutions to the quadratic equation 2x² - 3x - 2 = 0 are x = 2 and x = -1/2.
After obtaining the solutions using the quadratic formula, it's a good practice to verify whether these solutions indeed satisfy the original equation. This helps to ensure that there were no errors in the calculations and that the solutions are accurate. To verify, we substitute each solution back into the original equation, 2x² - 3x - 2 = 0, and check if the equation holds true.
Let's start with the first solution, x = 2. Substituting x = 2 into the equation, we get:
2(2)² - 3(2) - 2 = 2(4) - 6 - 2 = 8 - 6 - 2 = 0
Since the equation holds true when x = 2, this confirms that x = 2 is a valid solution.
Now, let's verify the second solution, x = -1/2. Substituting x = -1/2 into the equation, we get:
2(-1/2)² - 3(-1/2) - 2 = 2(1/4) + 3/2 - 2 = 1/2 + 3/2 - 2 = 4/2 - 2 = 2 - 2 = 0
Again, the equation holds true when x = -1/2, confirming that x = -1/2 is also a valid solution.
By verifying both solutions, we can confidently conclude that the solutions x = 2 and x = -1/2 are correct.
In this comprehensive guide, we have successfully solved the quadratic equation 2x² - 3x - 2 = 0 using the quadratic formula. We began by understanding the quadratic formula and its significance in solving quadratic equations. We then identified the coefficients a, b, and c from the given equation, ensuring accuracy in our substitutions. By systematically applying the quadratic formula, we obtained the solutions x = 2 and x = -1/2. To ensure the validity of our solutions, we verified them by substituting them back into the original equation, confirming that they indeed satisfy the equation.
The quadratic formula is a powerful tool that can be used to solve any quadratic equation, regardless of the nature of its roots. It provides a systematic approach to finding the solutions, making it an indispensable technique in mathematics. Understanding and applying the quadratic formula not only enhances your problem-solving skills but also provides a solid foundation for further exploration of algebraic concepts. By mastering this technique, you will be well-equipped to tackle a wide range of mathematical challenges and appreciate the elegance and versatility of algebra.
This step-by-step guide has provided a clear and concise explanation of how to use the quadratic formula. We encourage you to practice solving various quadratic equations to further solidify your understanding and proficiency. With consistent practice, you will become adept at applying the quadratic formula and confidently solving quadratic equations in any context.