Solving 8n² - 4n = 18 A Step-by-Step Guide
In the realm of mathematics, quadratic equations hold a significant position, appearing in various fields such as physics, engineering, and economics. Understanding how to solve these equations is crucial for anyone seeking to delve deeper into these disciplines. This article will provide a comprehensive guide on solving the quadratic equation 8n² - 4n = 18, breaking down each step and explaining the underlying principles.
Understanding Quadratic Equations
Before diving into the solution, it's essential to understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. In our case, the equation 8n² - 4n = 18 fits this definition, with 'n' being the variable and the coefficients being a = 8, b = -4, and c = -18 (after rearranging the equation).
Solving a quadratic equation means finding the values of the variable (in this case, 'n') that satisfy the equation. These values are also known as the roots or solutions of the equation. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. We will primarily focus on using the quadratic formula in this guide, as it is a versatile method applicable to all quadratic equations, regardless of whether they can be easily factored.
Rearranging the Equation
The first step in solving the equation 8n² - 4n = 18 is to rearrange it into the standard form ax² + bx + c = 0. To do this, we need to subtract 18 from both sides of the equation:
8n² - 4n - 18 = 0
Now, our equation is in the standard form, with a = 8, b = -4, and c = -18. This form is essential for applying the quadratic formula.
The Quadratic Formula
The quadratic formula is a powerful tool for finding the solutions of any quadratic equation. It states that for an equation in the form ax² + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b² - 4ac)) / 2a
In our case, the variable is 'n', so we will replace 'x' with 'n' in the formula. The formula provides two possible solutions, one using the plus sign (+) and the other using the minus sign (±). This reflects the fact that a quadratic equation can have up to two distinct real solutions.
Applying the Quadratic Formula to Our Equation
Now, let's apply the quadratic formula to our equation 8n² - 4n - 18 = 0. We have identified the coefficients as a = 8, b = -4, and c = -18. Plugging these values into the formula, we get:
n = (-(-4) ± √((-4)² - 4 * 8 * -18)) / (2 * 8)
Let's simplify this expression step by step.
Simplifying the Expression
First, we simplify the terms inside the parentheses and the square root:
n = (4 ± √(16 + 576)) / 16
Next, we add the numbers inside the square root:
n = (4 ± √592) / 16
Now, we need to simplify the square root of 592. We can find the prime factorization of 592 to help us simplify it. The prime factorization of 592 is 2⁴ * 37. Therefore:
√592 = √(2⁴ * 37) = 2²√37 = 4√37
Substituting this back into our equation, we get:
n = (4 ± 4√37) / 16
Now, we can factor out a 4 from the numerator:
n = 4(1 ± √37) / 16
Finally, we can simplify by dividing both the numerator and the denominator by 4:
n = (1 ± √37) / 4
The Two Solutions
Therefore, the two solutions for the equation 8n² - 4n = 18 are:
n₁ = (1 + √37) / 4
n₂ = (1 - √37) / 4
These are the exact solutions to the quadratic equation. We can also approximate these solutions using a calculator. The square root of 37 is approximately 6.083. Therefore:
n₁ ≈ (1 + 6.083) / 4 ≈ 7.083 / 4 ≈ 1.771
n₂ ≈ (1 - 6.083) / 4 ≈ -5.083 / 4 ≈ -1.271
So, the approximate solutions are n₁ ≈ 1.771 and n₂ ≈ -1.271.
Alternative Methods for Solving Quadratic Equations
While the quadratic formula is a reliable method, there are other techniques to solve quadratic equations, such as factoring and completing the square. Let's briefly discuss these methods.
Factoring
Factoring involves expressing the quadratic equation as a product of two linear factors. This method is efficient when the equation can be easily factored. However, not all quadratic equations can be factored using integers.
To factor the equation 8n² - 4n - 18 = 0, we first look for a common factor. In this case, we can divide the entire equation by 2:
4n² - 2n - 9 = 0
Now, we try to factor the quadratic expression. We are looking for two binomials (an + b)(cn + d) such that ac = 4, ad + bc = -2, and bd = -9. In this case, factoring is not straightforward, and the quadratic formula is a more suitable approach.
Completing the Square
Completing the square is another method for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial. While this method is less direct than the quadratic formula, it is valuable for understanding the structure of quadratic equations and is used in deriving the quadratic formula itself.
To solve 8n² - 4n - 18 = 0 by completing the square, we first divide the equation by 8 to make the coefficient of n² equal to 1:
n² - (1/2)n - (9/4) = 0
Next, we move the constant term to the right side of the equation:
n² - (1/2)n = 9/4
Now, we complete the square by adding the square of half the coefficient of n to both sides. The coefficient of n is -1/2, so half of it is -1/4, and the square of -1/4 is 1/16:
n² - (1/2)n + 1/16 = 9/4 + 1/16
n² - (1/2)n + 1/16 = 36/16 + 1/16
n² - (1/2)n + 1/16 = 37/16
The left side of the equation is now a perfect square trinomial, which can be factored as:
(n - 1/4)² = 37/16
Now, we take the square root of both sides:
n - 1/4 = ±√(37/16)
n - 1/4 = ±√37 / 4
Finally, we solve for n:
n = 1/4 ± √37 / 4
n = (1 ± √37) / 4
We arrive at the same solutions as we did using the quadratic formula.
Conclusion
In this article, we have explored how to solve the quadratic equation 8n² - 4n = 18 using the quadratic formula, factoring, and completing the square. The quadratic formula is a universal method that can be applied to any quadratic equation, while factoring and completing the square are useful techniques in specific cases. Understanding these methods provides a solid foundation for tackling more complex mathematical problems and real-world applications involving quadratic equations. Whether you are a student, engineer, or anyone interested in mathematics, mastering the solution of quadratic equations is an invaluable skill.
By following the steps outlined in this guide, you can confidently solve any quadratic equation you encounter. Remember to practice these techniques to solidify your understanding and develop your problem-solving skills. Quadratic equations are a fundamental part of mathematics, and their solutions play a crucial role in various scientific and engineering disciplines. Mastering these concepts will undoubtedly enhance your mathematical prowess and open doors to a deeper understanding of the world around you.
