Problem 9 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

(5) and (\sqrt{x}); 25 and x.

Step 1: Identify the Special Product

The given mathematical expression is \((5-\sqrt{x})^2\), which is a special product called a 'square of a binomial'. This can be expressed using the formula \((a-b)^2 = a^2 - 2ab + b^2\). In this problem, \(a = 5\) and \(b = \sqrt{x}\).

Step 2: Apply the Special Product Formula

Substitute \(a = 5\) and \(b = \sqrt{x}\) into the formula \((a-b)^2 = a^2 - 2ab + b^2\): \((5-\sqrt{x})^2 = (5)^2 - 2(5)(\sqrt{x}) + (\sqrt{x})^2\).

Step 3: Calculate Each Term

Now, calculate each term: - \((5)^2 = 25\) - \(2(5)(\sqrt{x}) = 10\sqrt{x}\) - \((\sqrt{x})^2 = x\)

Step 4: Fill in the Blanks

Using the calculated terms for substitution: The first blank, \(( ext{blank})^2\), is \((5)^2 = 25\). The second blank is for the product \(-2(5)(\sqrt{x}) = -10\sqrt{x}\). The third blank, \(( ext{blank})^2\), is \((\sqrt{x})^2 = x\).

Step 5: Rewrite the Equation

Fill in the remaining equation with the found values:\((5-\sqrt{x})^2 = \underbrace{25} - 10\sqrt{x} + \underbrace{x}\). Copy the original blanks with the filled values for conciseness:\((5-\sqrt{x})^{2} = (5)^{2} - 2(5)(\sqrt{x}) + (\sqrt{x})^{2}\).

Key Concepts

Square of a Binomial

The square of a binomial refers to squaring an expression made of two terms, like \((a-b)^2\) or \((a+b)^2\). Essentially, it means multiplying the binomial by itself. To simplify a squared binomial, we expand it using the formula for square of a binomial: \[(a \pm b)^2 = a^2 \pm 2ab + b^2\]This formula allows us to break down the squared binomial into specific, easier to handle terms.For example, when you have \((5-\sqrt{x})^2\), this can be expanded as:

• The first term squared: \(a^2 = 5^2 = 25\).
• Twice the product of the two terms: \(-2(5)(\sqrt{x}) = -10\sqrt{x}\).
• The second term squared: \((\sqrt{x})^2 = x\).

Once expanded, you have an easier expression to work with that highlights each component clearly.

Binomial Expansion

Binomial expansion refers to expressing a binomial raised to any power in an expanded form. The process allows us to transform an expression like \((a-b)^n\) or \((a+b)^n\) into a more usable polynomial form.To perform binomial expansion specifically for squaring, as in the formula \((a-b)^2\), we expand using:

• The square of the first term, \(a^2\).
• The double product of both terms, \(-2ab\).
• The square of the second term, \(b^2\).

For instance, for \((5-\sqrt{x})^2\), we substitute:

• \(5\) as \(a\) and \(\sqrt{x}\) as \(b\).
• Calculate \(a^2 = 25\), \(-2ab = -10\sqrt{x}\), and \(b^2 = x\).

The expanded form clearly shows each constituent part of the original binomial amplified, offering an understanding of the relationship between the original binomial terms.

Algebraic Expressions

Algebraic expressions consist of variables and constants combined using mathematical operations. Understanding them is crucial as they form the basis of algebra.An algebraic expression might include terms such as integers, fractions, and square roots, combined using operations like addition and subtraction. With binomials such as \((5-\sqrt{x})\), algebraic expressions take the shape of sums and differences of terms.In the context of special products like \((5-\sqrt{x})^2\), every term in the expanded expression represents an algebraic entity:

• \(25\) is a constant term.
• \(-10\sqrt{x}\) is a term with both a constant factor and a variable factor.
• \(x\) represents a variable term.

Understanding algebraic expressions in these terms allows us to solve and simplify expressions precisely, ensuring that complex operations remain manageable.