Problem 69 - Algebra 2

Question

Simplify each expression. $$ \frac{-2-2 \sqrt{5}}{1-\sqrt{5}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\frac{-2-2 \sqrt{5}}{1-\sqrt{5}}\) is \(3 + \sqrt{5}\)

Step 1: Identify the Quadratic Formula

From the expression, you can identify a, b and equivalently the quadratic roots. The fraction's denominator is in the form of a difference of two squares \((a-b)\), where \(a = 1\) and \(b = \sqrt{5}\).

Step 2: Rationalize the Denominator

To rationalize the denominator, multiply both top and bottom of the fraction by the conjugate of the denominator, which is \((1+\sqrt{5})\). Doing this will make the denominator a rational number: \(\frac{-2-2\sqrt{5}}{1-\sqrt{5}} = \frac{-2-2\sqrt{5}}{1-\sqrt{5}} * \frac{1+\sqrt{5}}{1+\sqrt{5}}\)

Step 3: Simplify the Expression

Multiply out the numerator and denominator: Numerator: \((-2-2\sqrt{5})(1+\sqrt{5}) = -2 - 2\sqrt{5} -2\sqrt{5} -10 = -12 - 4\sqrt{5}\). Denominator: \((1-\sqrt{5})(1+\sqrt{5}) = 1^2 - (\sqrt{5})^2 = 1 - 5 = -4\). So, the expression becomes \(\frac{-12 - 4\sqrt{5}}{-4}\). When -4 divides both -12 and -4\sqrt{5}, it simplifies to \(3+ \sqrt{5}\).

Step 4: Check the Result

Double-check this result by computing \((3 + \sqrt{5})(1 - \sqrt{5})\) to make sure it gives \(-2 - 2\sqrt{5}\). Both checks out, so \(3+ \sqrt{5}\) is indeed the simplified form of the expression

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operations. In essence, they are a combination of constants and variables linked together by operations such as addition, subtraction, multiplication, and division. For instance, in the expression \(-2 - 2\sqrt{5}\), both numbers \(-2\) and the product \(-2\sqrt{5}\) are considered terms.

When dealing with fractions, algebraic expressions can sometimes appear in the numerator or the denominator. This might make it necessary to simplify the expression, especially when roots are involved. Simplifying these kinds of fractions often involves techniques that can resolve irrational or complex numbers in the denominator, clearing up the expression significantly. It's crucial to understand these components to tackle more intricate algebraic problems effectively.

Working with algebraic expressions is an important skill in algebra as it forms the basis for expressing equations and inequalities. It also provides a foundation for more advanced topics like derivatives and integrals in calculus.

Conjugates

Conjugates are pairs of expressions formed by changing the sign between two terms. For example, if we have an expression like \(1 - \sqrt{5}\), its conjugate is \(1 + \sqrt{5}\). Conjugates are particularly useful in rationalizing denominators, which is a fundamental process in mathematics to eliminate square roots or complex numbers from the denominator of a fraction.

The main idea here is that when you multiply a number by its conjugate, the result is a difference of squares, which is a rational number. This can be seen with \((1-\sqrt{5})(1+\sqrt{5})\), which simplifies to \(1^2 - (\sqrt{5})^2 = 1 - 5 = -4\). By multiplying both the numerator and the denominator by the conjugate, you can simplify expressions to a more workable form.

In the given exercise, using conjugates allowed the transformation of \(1-\sqrt{5}\) in the denominator into \(-4\), which can then be easily managed using basic arithmetic operations. Simplifying expressions using conjugates is both a necessary algebraic technique and a powerful tool to achieve simplified forms.

Simplifying Expressions

Simplifying expressions is a key process that aims to make a complex expression easier to work with. This process generally involves arithmetic operations, expanding, factoring, and canceling common factors. For the exercise given, after rationalizing the denominator, the expression transforms into \(\frac{-12 - 4\sqrt{5}}{-4}\).

To simplify this, you divide each term in the numerator by the value in the denominator, \(-4\). This results in simplifying to \(3 + \sqrt{5}\). Simplification significantly reduces the complexity of an expression. In doing so, we're essentially expressing the same value in a more straightforward, cleaner form.

Methods like factoring, canceling out terms, and simplifying radicals come in handy. This not only makes equations more manageable but research also shows that simplified expressions help in computational efficiency, which is essential in solving advanced mathematical problems effectively. Therefore, mastering the art of simplifying expressions is crucial for both academic progress in mathematics and practical real-life applications.