Problem 44 - College Algebra

Question

Factor the difference of two squares. $$ 36 x^{2}-49 y^{2} $$

Step-by-Step Solution

Answer

The factorized form of the equation \(36x^2 - 49y^2\) is \((6x - 7y)(6x + 7y)\).

Step 1: Understanding the Equation

We have the expression \(36x^2 - 49y^2\), This is a difference of squares equation because we have subtracted the square of one term from the square of another.

Step 2: Identify 'a' and 'b'

The terms 'a' and 'b' in the formula can be replaced by the two expressions that are being squared. Here, 'a' is \(6x\) because \((6x)^2 = 36x^2\) and 'b' is \(7y\) because \((7y)^2 = 49y^2\). So, after identifying, we have \(a = 6x\) and \(b = 7y\).

Step 3: Apply the Difference Of Two Squares Formula

Now we need to substitute 'a' and 'b' into our difference of two squares formula: \(a^2 - b^2 = (a - b)(a + b)\). On substituting 'a' and 'b' we get: \((6x)^2 - (7y)^2 = (6x - 7y)(6x + 7y)\).

Step 4: Simplify the Factorized Form

Finally, simplify the expressions, and the factorized form would be: \(36x^2 - 49y^2 = (6x - 7y)(6x + 7y)\).

Key Concepts

Factoring

Factoring plays a crucial role in algebra, particularly when dealing with polynomials. It involves breaking down a complex expression into simpler factors that, when multiplied together, give the original expression. This process makes solving equations easier and more straightforward.

In the case of a difference of squares, like in the expression given in the exercise, factoring is approached by recognizing that the expression fits a particular form:

• It must have two terms.
• Each term must be a perfect square.
• There should be a subtraction (difference) between these terms.

Factoring the difference of squares is straightforward once you identify the squared terms. Using the formula \[a^2 - b^2 = (a - b)(a + b)\], you replace \(a\) and \(b\) with the terms you identified. This method simplifies the expression significantly.

Algebraic Expressions

Algebraic expressions form the backbone of algebra and involve combinations of numbers, variables, and operations. Understanding these expressions is key to manipulating equations and finding solutions.

A typical algebraic expression could look like \(36x^2 - 49y^2\), which combines numbers (36, 49), variables \(x\) and \(y\), and an operation (subtraction). Each component of an algebraic expression carries meaning:

• **Variables** like \(x\) and \(y\) represent numbers that can vary throughout calculations.
• **Coefficients** such as 36 and 49 indicate how many times a variable is multiplied by itself (in squared terms).
• **Constants** may also appear as standalone numbers added or subtracted within the expression.

Successfully working with algebraic expressions means recognizing these components and understanding how they interact within operations.

Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, combined using only addition, subtraction, and multiplication. They are named based on the highest degree of their terms.

The polynomial in the exercise, \(36x^2 - 49y^2\), is a quadratic polynomial because the highest degree of each variable term is 2 (\(x^2\) and \(y^2\)).

• **Terms**: Each is a part of the polynomial, separated by + or - sign. For \(36x^2 - 49y^2\), these are \(36x^2\) and \(-49y^2\).
• **Coefficients**: Numbers in front of the variables, such as 36 and -49.
• **Degree**: The highest power of the variable within the expression, which is 2 in this case.

Understanding how these elements form polynomials allows you to categorize and solve them using relevant techniques, such as factoring, in this example.