Problem 9 - Intermediate Algebra

Question

Fill in the blanks to complete this special product: $$ (5-\sqrt{x})^{2}=(\quad)^{2}-\quad(5)(\sqrt{x})+(\quad)^{2} $$ $$ =\quad-10 \sqrt{x}+ $$

Step-by-Step Solution

Answer

\((5)^2=25\), \(-2(5)(\sqrt{x})\), \((\sqrt{x})^2=x\), so final result is \(25 - 10\sqrt{x} + x\).

Step 1: Write Down the Formula for Squaring a Binomial

The formula for squaring a binomial \((a-b)^2\) is \(a^2 - 2ab + b^2\). In this problem, \(a=5\) and \(b=\sqrt{x}\).

Step 2: Apply the Formula to the Given Expression

First, square the first term: \(5^2 = 25\). Next, use the middle term formula \(-2ab\), which gives us \(-2 \times 5 \times \sqrt{x} = -10\sqrt{x}\). Lastly, square the second term: \((\sqrt{x})^2 = x\).

Step 3: Substitute Values into the Expression

Using the values calculated from Step 2, substitute them back into the expression: 1. \((5)^2 = 25\)2. \(-2 \times 5 \times \sqrt{x} = -10\sqrt{x}\)3. \((\sqrt{x})^2 = x\)So, \((5-\sqrt{x})^2 = 25 - 10\sqrt{x} + x\).

Step 4: Identify the Blanks in the Original Expression

The blanks are filled as follows:1. The first blank \((\quad)^2\) is \(5\), so \(5^2\).2. The second blank in \(-\quad(5)(\sqrt{x})\) is \(2\), so \(-2(5)(\sqrt{x})\).3. The third blank \((\quad)^2\) is \(\sqrt{x}\), so \((\sqrt{x})^2\).

Key Concepts

Understanding the Binomial Theorem

The binomial theorem is a significant topic in algebra. It allows us to expand binomials raised to any positive integer power. At its core, this theorem provides a formula to describe the expansion of powers of binomials, thus simplifying what might initially seem complex. The theorem states: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^{k}\]where \(\binom{n}{k}\) is a binomial coefficient computed as \(\frac{n!}{k!(n-k)!}\). This coefficient indicates how many ways you can pick \(k\) items from \(n\) without regard to order.
By applying the binomial theorem, calculations that seem complicated at first become methodical and straightforward. Since each term in the expansion of \((a+b)^2\), for example, involves a combination of powers of \(a\) and \(b\), knowing this theorem is incredibly useful when working with algebraic expressions and expanding them efficiently.

Squaring a Binomial

Squaring a binomial involves expanding an expression of the form \((a - b)^2\). It's one of the simplest forms of applying the binomial theorem. The formula to remember is:\[a^2 - 2ab + b^2\]This formula gives us the square of the first term, minus twice the product of both terms, plus the square of the second term.
For instance, let's take a binomial like \((5 - \sqrt{x})^2\). Here, \(a=5\) and \(b=\sqrt{x}\). To square it, we do the following steps:

• Square the first term: \(5^2 = 25\)
• Multiply the two terms and double it for the middle term: \(-2 \times 5 \times \sqrt{x} = -10\sqrt{x}\)
• Square the second term: \((\sqrt{x})^2 = x\)

Putting it together, \((5 - \sqrt{x})^2 = 25 - 10\sqrt{x} + x\). By breaking it down, each step ties back into the original formula, making binomial square expansions easier to handle.

Algebraic Expressions: Unpacking the Basics

Algebraic expressions form the backbone of algebra. They comprise numbers, variables, and operations (such as addition, subtraction, and multiplication) put together to form meaningful mathematical statements.
For example, a simple algebraic expression like \(5x + 3\) includes a variable \(x\) and constants 5 and 3. Variables stand in for unknown values, and they can change depending on the context or the problem being solved.
When dealing with more complex expressions like the one involving binomials \((5 - \sqrt{x})^2\), algebraic expressions become essential in arranging, simplifying, and solving for unknowns. Expressions can be manipulated using various algebraic rules and operations to find solutions or rewrite them in an equivalent, often simpler, form.
Grasping how to work with these expressions is crucial. It opens doors to understanding more advanced algebraic concepts and solving equations that arise in both academic exercises and real-world applications.