Problem 20 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ (5 h+2 k)(5 h-2 k) $$

Step-by-Step Solution

Answer

Question: Simplify the given algebraic expression: \((5h + 2k)(5h - 2k)\) Answer: The simplified expression for the given algebraic equation is $$25h^2 - 4k^2$$.

Step 1: Identify the terms and apply binomial product rule

The given expression is \((5h + 2k)(5h - 2k)\). We will apply the binomial product rule, where \(a = 5h\) and \(b = 2k\). According to the formula, \((a + b)(a - b) = a^2 - b^2\). So, let's find \(a^2\) and \(b^2\).

Step 2: Calculate \(a^2\)

To find \(a^2\), we will square \(a\), which is \(5h\). So, \(a^2 = (5h)^2\). Now, we will calculate the square of \(5h\): \(a^2 = 25h^2\).

Step 3: Calculate \(b^2\)

To find \(b^2\), we will square \(b\), which is \(2k\). So, \(b^2 = (2k)^2\). Now, we will calculate the square of \(2k\): \(b^2 = 4k^2\).

Step 4: Apply the formula \((a + b)(a - b) = a^2 - b^2\)

We've found that \(a^2 = 25h^2\) and \(b^2 = 4k^2\). Now, we can apply the formula \((a + b)(a - b) = a^2 - b^2\) and substitute the values: \((5h + 2k)(5h - 2k) = 25h^2 - 4k^2\).

Step 5: Final simplified expression

The simplified expression of the given algebraic equation is: $$25h^2 - 4k^2$$

Key Concepts

Binomial Product Rule

The binomial product rule is a handy algebraic tool for simplifying certain expressions, especially those represented as a product of two binomials in the form \((a + b)(a - b)\). This special structure allows us to apply a direct formula:

• \((a + b)(a - b) = a^2 - b^2\)

This is called the difference of squares formula.
In this formula, two symmetrical expressions multiplied together simplify into the subtraction of squares of their respective terms.
In the given exercise, the expression \((5h + 2k)(5h - 2k)\) is recognized as a product of such binomials.
By identifying \(a = 5h\) and \(b = 2k\), we can directly apply this rule to efficiently simplify more complex algebraic problems.Understanding this can save a lot of time compared to expanding and then gathering like terms.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form without changing their value.
After identifying and applying the appropriate algebraic rules, expressions often become much more manageable.
In the context of the exercise, the simplification process using the binomial product rule follows a quick and structured pathway.
Initially, we might see the full expanded terms if we multiply separately: - \((5h + 2k)(5h - 2k)\) expands to - \((5h)(5h) - (5h)(2k) + (2k)(5h) - (2k)(2k)\)However, by identifying the expression’s form and applying the difference of squares, we go directly to \(25h^2 - 4k^2\).
This is why learning how to simplify with known formulas is so efficient. Simplification is not merely about reducing; it aids in understanding the structure and behavior of algebraic expressions.

Squaring Terms

Squaring a term means multiplying the term by itself.
It’s a fundamental operation in algebra, often encountered when using formulas, such as the binomial product rule. Understanding squaring is crucial because it prepares us to handle more complex algebraic expressions and equations. For example, in the expression \((5h)^2\), squaring involves:

• Multiplying \(5h \times 5h\)
• Resulting in \(25h^2\)

This shows both the increase in the coefficient (from 5 squared becomes 25) and the new exponent attached to the variable \(h\).
Similarly, for \((2k)^2\), this results in \(4k^2\).Understanding squaring is an essential building block in managing equations and effectively applying algebraic rules, leading to accurately simplified results.