Problem 73 - Intermediate Algebra

Question

Multiply and then simplify if possible. $$ (\sqrt{x-1}+5)^{2} $$

Step-by-Step Solution

Answer

\((\sqrt{x-1} + 5)^2 = x + 24 + 10\sqrt{x-1}\).

Step 1: Understand the Problem

The problem requires us to square the binomial \((\sqrt{x-1} + 5)\) and then simplify the resulting expression if possible.

Step 2: Apply the Binomial Formula

To expand the expression \((\sqrt{x-1} + 5)^2\), we use the binomial formula: \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = \sqrt{x-1}\) and \(b = 5\).

Step 3: Calculate Each Term

First, calculate \(a^2\): \[(\sqrt{x-1})^2 = x - 1\]. Next, calculate \(2ab\): \[2 \cdot \sqrt{x-1} \cdot 5 = 10\sqrt{x-1}\]. Finally, calculate \(b^2\): \[5^2 = 25\].

Step 4: Combine the Terms

Add the calculated terms together: \[(\sqrt{x-1})^2 + 2(\sqrt{x-1})(5) + 5^2 = x - 1 + 10\sqrt{x-1} + 25\].

Step 5: Simplify the Expression

Combine the constant terms: \[x - 1 + 25 = x + 24\]. So, the expression becomes: \[x + 24 + 10\sqrt{x-1}\].This is the simplified form.

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form while preserving their original value. This process helps in making expressions easier to work with or understand. Let's dive into some key approaches used when simplifying:

• Combine Like Terms: Gather terms with the same variable and exponent. For example, combining terms like 3x and 2x results in 5x.
• Perform Arithmetic: Calculate any arithmetic within the expression, such as adding or subtracting constants. In our exercise, from the calculation, the expression \(x - 1 + 25\) simplifies to \(x + 24\).
• Factor When Possible: Sometimes expressions can be factored to simplify them further, although our specific exercise doesn't require additional factoring.

Simplification ensures that algebraic expressions are easy to interpret and evaluate, aiding us in further mathematical operations.

Exponents and Radicals

Understanding exponents and radicals is crucial as they are foundational in algebra.
Exponents indicate repeated multiplication of a number, while radicals denote roots.

• Exponents: An expression like \(a^2\) means \(a\) multiplied by itself. It reflects the 'power' to which the base number is raised.
• Radicals: The square root radical, \(\sqrt{}\), commonly arises in mathematics. It signifies the number that, when multiplied by itself, gives the original number inside the radical.For instance, \(\sqrt{9} = 3\), because \(3 \times 3 = 9\).

In the exercise, we handled the radical by squaring \((\sqrt{x-1})^2\). This allowed the radical and exponent to 'cancel out,' resulting in \(x - 1\). Knowing how radicals and exponents function can demystify and streamline complex algebraic manipulations.

Polynomial Multiplication

Polynomial multiplication is a method used to multiply expressions involving terms with variables. It's a common procedure in algebra that helps in expanding expressions. The most relevant method for this operation is applying the binomial expansion.

• Binomial Theorem: This formula \((a + b)^2 = a^2 + 2ab + b^2\) simplifies the multiplication of binomials. The terms \(a\) and \(b\) are multiplied according to this pattern to give a neat expanded form.
• Multiply Each Term Individually: In the exercise, the binomial \((\sqrt{x-1} + 5)^2\) used this principle. Each component—\((\sqrt{x-1})^2\), \(2(\sqrt{x-1})(5)\), and \(5^2\)—was calculated individually.

Multiplying polynomials requires attention to detail, but by following standardized formulas such as the binomial theorem, complex expressions become easier to manage and solve.