Problem 35 - Intermediate Algebra

Question

Write the conjugate of each expression. $$ \sqrt{2}+x $$

Step-by-Step Solution

Answer

The conjugate is \(\sqrt{2} - x\).

Step 1: Understanding Conjugate Roots

When you have a two-term expression involving a square root, such as \(\sqrt{2} + x\), the conjugate is formed by changing the sign between the two terms. The conjugate of \(\sqrt{2} + x\) would be \(\sqrt{2} - x\).

Step 2: Verify the Conjugate

Multiply the original expression and its conjugate to check the operation. Multiply \((\sqrt{2} + x)(\sqrt{2} - x)\). Use the difference of squares formula: \((a+b)(a-b) = a^2 - b^2\).

Step 3: Apply the Formula

Identify \(a = \sqrt{2}\) and \(b = x\), then apply the formula: \((\sqrt{2})^2 - x^2\). This simplifies to: \(2 - x^2\). This confirms that \(\sqrt{2} - x\) is indeed the conjugate.

Key Concepts

Difference of Squares

The difference of squares is a fundamental algebraic expression with a specific pattern. It describes the result when you multiply two conjugate pairs. A standard form is: \((a+b)(a-b)\), resulting in \(a^2 - b^2\). In our example, we have \(a = \sqrt{2}\) and \(b = x\). The conjugate of the expression \(\sqrt{2} + x\) is \(\sqrt{2} - x\).

• Recognize the pattern: the expression fits a \(\text{sum and difference}\) format, which is ideal for the difference of squares method.
• Apply the formula: multiply these expressions to eliminate square roots and verify your results.

Thus, multiplying \((\sqrt{2} + x)(\sqrt{2} - x)\) yields \(\sqrt{2}^2 - x^2 = 2 - x^2\). This is the simplified form without any square roots in it.

Square Roots in Expressions

Square roots can add complexity to algebraic expressions, but understanding how they interact with other terms makes handling them easier.

• Square roots are the opposite of squares. For example, \(\sqrt{2}\) means a number which, when squared, equals 2.
• They often appear in expressions under radicals, and they can lead to non-rational numbers.

In expressions like \(\sqrt{2} + x\), the challenge is in simplifying or transforming them, often by using their conjugates. By forming and using the conjugate, \(\sqrt{2} - x\), one can eliminate the square root, simplifying algebraic investigation and calculation.

Verifying Expressions

Verifying expressions is crucial in algebra to ensure calculations are correct and transformations are accurate. Let's take the expression \(\sqrt{2} + x\) and its conjugate \(\sqrt{2} - x\).

• Multiply them to check: this uses the difference of squares concept, ensuring that transformations are accurate.
• Result: when you multiply, \((\sqrt{2} + x)(\sqrt{2} - x) = 2 - x^2\), confirms that the operation and the chosen conjugate simplify correctly.

By verifying expressions, we confirm accuracy in mathematical transformations. It helps discover errors and understand how algebraic structures behave under operations.