Solving (√3 - 1)/(√3 + 1) = A + B√3 Find A And B
In this article, we delve into the process of finding the values of a and b in the equation (√3 - 1)/(√3 + 1) = a + b√3. This type of problem is a classic example of rationalizing the denominator and then equating coefficients. We will break down each step involved in solving this equation, ensuring a clear and comprehensive understanding. The goal is to manipulate the left-hand side of the equation to match the form of the right-hand side, allowing us to directly identify the values of a and b. This involves algebraic manipulation, a key skill in mathematics, particularly in algebra and calculus. This topic is not only fundamental for high school students but also crucial for those pursuing higher education in science and engineering fields. Mastering such problems enhances analytical and problem-solving skills, which are essential for various applications in real-world scenarios. We aim to provide a step-by-step solution, making it easier for students and enthusiasts to grasp the underlying concepts and techniques. The ability to solve such equations is a stepping stone to more complex mathematical challenges, including those encountered in advanced courses and competitive exams. This article will serve as a valuable resource for anyone looking to improve their algebraic manipulation skills and deepen their understanding of irrational numbers and their properties. We will also discuss the common mistakes students make and provide tips to avoid them, ensuring a thorough understanding of the process. By the end of this article, readers should be able to confidently solve similar problems and apply these techniques to other areas of mathematics.
Rationalizing the Denominator
To solve the equation (√3 - 1)/(√3 + 1) = a + b√3, the first step is to rationalize the denominator of the fraction on the left-hand side. Rationalizing the denominator means eliminating the square root from the denominator. To do this, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of √3 + 1 is √3 - 1. This technique is crucial because it allows us to transform an expression with an irrational denominator into an equivalent expression with a rational denominator, making it easier to compare and manipulate. Multiplying by the conjugate is a standard algebraic technique used to simplify expressions involving square roots. This step is essential for isolating the terms and eventually solving for a and b. The conjugate method works because when you multiply an expression by its conjugate, the middle terms cancel out, leaving you with a rational number in the denominator. This process not only simplifies the expression but also allows us to express it in a form that is easier to compare with the right-hand side of the equation. Understanding and mastering this technique is fundamental in algebra and will be helpful in more advanced mathematical problems. Rationalizing the denominator is not just a mathematical trick; it’s a powerful tool that helps simplify complex expressions and makes them more manageable. In many areas of mathematics and science, simplifying expressions is a crucial step in problem-solving, and rationalizing the denominator is a key technique to achieve this. By carefully applying this method, we can transform the given equation into a form that allows us to easily identify the values of a and b, which is the ultimate goal of this problem.
Multiplying by the Conjugate
Multiplying both the numerator and the denominator by the conjugate (√3 - 1), we get:
(√3 - 1) / (√3 + 1) * (√3 - 1) / (√3 - 1)
This step is the heart of rationalizing the denominator. By multiplying both the numerator and denominator by the same expression, we are essentially multiplying by 1, which doesn't change the value of the fraction but changes its form. The key here is that multiplying by the conjugate will eliminate the square root in the denominator. This process transforms the denominator into a rational number, which simplifies the entire expression. When we multiply the numerators, we get (√3 - 1)(√3 - 1), which is equivalent to (√3 - 1)^2. When we multiply the denominators, we get (√3 + 1)(√3 - 1), which is a difference of squares. Understanding the properties of conjugates and the difference of squares is crucial for this step. The difference of squares identity, which states that (a + b)(a - b) = a^2 - b^2, is particularly useful here. Applying this identity makes the simplification process much more straightforward. The goal of this multiplication is to transform the given fraction into a form that is easier to compare with the right-hand side of the equation (a + b√3). By eliminating the square root in the denominator, we pave the way for a direct comparison of coefficients, allowing us to solve for a and b. This step demonstrates a fundamental principle in algebra: manipulating expressions to reveal underlying structures and simplify calculations. Mastering this technique is essential for success in more advanced mathematical studies.
Expanding the Numerator and Denominator
Now, let's expand the numerator and the denominator:
Numerator:
(√3 - 1)(√3 - 1) = (√3)^2 - 2(√3)(1) + (1)^2 = 3 - 2√3 + 1 = 4 - 2√3
Denominator:
(√3 + 1)(√3 - 1) = (√3)^2 - (1)^2 = 3 - 1 = 2
Expanding the numerator involves applying the binomial square formula, which is a fundamental concept in algebra. The binomial square formula states that (a - b)^2 = a^2 - 2ab + b^2. In this case, a is √3 and b is 1. Applying this formula correctly is crucial for obtaining the correct result. The term (√3)^2 simplifies to 3 because the square root and the square cancel each other out. The term -2(√3)(1) simplifies to -2√3, and the term (1)^2 simplifies to 1. Combining these terms gives us the expression 4 - 2√3. Expanding the denominator utilizes the difference of squares formula, which is another essential algebraic identity. This formula states that (a + b)(a - b) = a^2 - b^2. In this case, a is √3 and b is 1. Applying this formula simplifies the denominator to (√3)^2 - (1)^2, which further simplifies to 3 - 1, resulting in 2. The difference of squares formula is a powerful tool for simplifying expressions and is widely used in various areas of mathematics. Both the numerator and denominator expansions are key steps in simplifying the original expression. These expansions transform the expression into a more manageable form, making it easier to identify the values of a and b. Accurate application of algebraic formulas and careful calculation are essential to avoid errors in this process.
Simplifying the Fraction
So, the fraction becomes:
(4 - 2√3) / 2
We can simplify this fraction by dividing both terms in the numerator by the denominator:
4/2 - (2√3)/2 = 2 - √3
Simplifying the fraction involves dividing each term in the numerator by the denominator. This step is a crucial part of reducing the expression to its simplest form. In this case, we have (4 - 2√3) / 2. We divide both the constant term 4 and the term with the square root, -2√3, by the denominator 2. Dividing 4 by 2 gives us 2. Dividing -2√3 by 2 gives us -√3. Therefore, the simplified expression is 2 - √3. This simplification is essential because it brings the expression closer to the form a + b√3, which allows us to directly compare the coefficients and find the values of a and b. This step demonstrates the importance of basic arithmetic operations in simplifying algebraic expressions. Dividing by the common factor not only makes the expression simpler but also reveals the underlying structure more clearly. Simplifying fractions is a fundamental skill in mathematics, and mastering this skill is crucial for success in algebra and beyond. This step prepares the expression for the final comparison, which will lead us to the solution for a and b. The ability to simplify fractions efficiently is a key component of mathematical fluency.
Equating Coefficients
Now we have:
2 - √3 = a + b√3
By comparing the terms, we can see that:
a = 2
b = -1
Equating coefficients is the final step in solving for a and b. This method is based on the principle that if two expressions of the form a + b√3 are equal, then their corresponding coefficients must be equal. In other words, the constant terms must be equal, and the coefficients of the √3 terms must be equal. In our case, we have the equation 2 - √3 = a + b√3. By comparing the constant terms, we can directly see that a = 2. The constant term on the left-hand side is 2, and the constant term on the right-hand side is a. By comparing the coefficients of the √3 terms, we can see that b = -1. The coefficient of √3 on the left-hand side is -1, and the coefficient of √3 on the right-hand side is b. This method of equating coefficients is a powerful technique for solving equations where irrational numbers are involved. It allows us to break down the equation into simpler parts and solve for the unknowns directly. The accuracy of this method relies on the correct simplification and transformation of the original equation. Equating coefficients is not just a mathematical trick; it’s a logical deduction based on the properties of numbers and equality. This technique is widely used in various areas of mathematics, including algebra, calculus, and complex numbers. The ability to equate coefficients efficiently is a key skill for solving mathematical problems involving irrational numbers and algebraic expressions. By correctly identifying and comparing the coefficients, we arrive at the final solution for a and b.
Conclusion
Therefore, the values of a and b are a = 2 and b = -1. This problem demonstrates the importance of rationalizing the denominator and equating coefficients to solve equations involving irrational numbers. Mastering these techniques is crucial for success in algebra and beyond.
Rationalizing the denominator, equating coefficients, algebraic manipulation, irrational numbers, conjugate, difference of squares, binomial square formula, simplification, equation solving, square roots, mathematics.
