Problem 24 - Algebra 1: Concepts and Skills

Question

Factor the expression. $$ 36 x^{2}+25 $$

Step-by-Step Solution

Answer

The factored form of the expression \(36x^2 + 25\) is \((6x+5i)(6x-5i)\).

Step 1: Establish the 'a' and 'b' values

In the given equation \(36x^2 + 25\), we can establish our 'a' and 'b' values. 'a' is equal to \(6x\) (since \((6x)^2 = 36x^2\)) and 'b' is equal to 5 (since \(5^2 = 25\))

Step 2: Plug the values into the factoring formula

Next, plug 'a' and 'b' values into the \(a^{2} + b^{2} = (a+b\sqrt{-1})(a-b\sqrt{-1})\) formula. This gives \((6x+5\sqrt{-1})(6x-5\sqrt{-1})\)

Step 3: Simplify

Finally, simplify \(\sqrt{-1}\) to 'i', an imaginary unit. Therefore, the factored form of \(36x^2 + 25\) is: \((6x+5i)(6x-5i)\)

Key Concepts

Imaginary Unit

When working with square roots of negative numbers, we make use of the imaginary unit, denoted as 'i'. This creation in mathematics allows us to extend the real number line to include solutions to equations that would otherwise have no solution. The key relationship at the heart of this unit is that \( i^2 = -1 \).

Understanding this concept is critical when dealing with quadratic equations that have negative discriminants, as it enables us to factor expressions that involve the square root of a negative number. The given exercise involves the sum of squares—specifically, \( 36x^2 + 25 \). In this case, there isn't a direct way to factor the expression using real numbers alone, but using the imaginary unit 'i', we can express the solution. It is vital to be comfortable with operations involving 'i' to maneuver through advanced algebra and complex number systems.

Difference of Squares

A fundamental technique in algebra is recognizing patterns in expressions, one of which is the difference of squares. This pattern takes the form \( a^2 - b^2 \) and factors into \( (a + b)(a - b) \). It's one of the first factoring techniques taught, as it's vital in simplifying many algebraic expressions.

However, when one encounters a sum of squares, such as in our exercise, the expression doesn't factor in a similar fashion within the real numbers. Instead, when expressed with the imaginary unit, the sum of squares can be thought of as a difference of squares where one of the squares is negative. In our step by step solution, the sum \( 36x^2 + 25 \) is treated like a difference by incorporating 'i', enabling the expression to be factored. This process is essential when solving equations in complex numbers, further highlighting the importance of understanding this algebraic property.

Algebraic Expressions

The world of algebra is filled with algebraic expressions, which are combinations of numbers, variables, and arithmetic operations. Grasping how to work with these expressions is crucial for students. It allows them to solve equations, simplify expressions, and understand the relationships between variables. In the provided exercise, the expression \( 36x^2 + 25 \) is considered an algebraic expression because it includes variables (\( x \)), constants (36 and 25), and the operations of exponentiation and addition.

When factoring algebraic expressions like the one in the exercise, it is essential to identify common patterns or structures, such as a difference of squares or perfect square trinomials. While the exercise itself does not directly factor into real numerical components, the use of the imaginary unit extends the real number system to allow for the factoring of such expressions, underlining the significance of comprehending these algebraic concepts within broader mathematical contexts.