Problem 45 - Algebra 1: Concepts and Skills

Question

Factor the expression. $$ x^{2}+12 x+36 $$

Step-by-Step Solution

Answer

The factored form of the expression \( x^{2} + 12x + 36 \) is \( (x + 6)^2 \).

Step 1: Identify the coefficients

First, identify the coefficients for the quadratic equation \( x^{2} + 12x + 36 \). The coefficients are a=1, b=12, and c=36.

Step 2: Determine the factors of the constant term

Find two numbers that multiply to 36 (the constant term, or 'c') and add to 12 (the coefficient of 'x', or 'b'). In this case, the numbers are 6 and 6, because 6*6 = 36 and 6+6 = 12.

Step 3: Write the factored form

Now, write the expression in its factored form using the numbers obtained in the previous step. The expression \( x^{2} + 12x + 36 \) factors to \( (x + 6)(x + 6) \) or \( (x + 6)^2 \).

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and represent expressions set to the second degree. These are equations that can be generally expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients and \( x \) is the variable. The coefficient \( a \) is never zero, as that would make it a linear equation instead.
Quadratic equations often appear in problems that describe area, motion, and many other natural phenomena. Solving these equations helps us find the values of \( x \) that make the equation true, known as the "roots".
Understanding the behavior of quadratic equations is important as they usually have a graph in the shape of a parabola. The parabola can open upwards or downwards depending on the sign of \( a \). A positive \( a \) opens upwards while a negative \( a \) opens downwards, resembling a frown.
In the case of our specific exercise, \( x^2 + 12x + 36 \) is the quadratic equation we are working with. Here, the aim is to solve by expressing as a product of two binomials.

Factoring Techniques

Factoring is a key technique used to simplify quadratic equations. It involves breaking down the equation into the product of simpler expressions.
To factor a quadratic equation like \( x^2 + 12x + 36 \), you should look for two numbers that both multiply to the constant term \( c = 36 \), and add up to the linear coefficient \( b = 12 \). These numbers are crucial because they determine how the quadratic will split into two binomials.
For our exercise, those numbers are 6 and 6. This means we can factor the quadratic as \( (x+6)(x+6) \) or more simply as \( (x+6)^2 \). This form reveals that \( x = -6 \) is a root repeated twice, also known as a "double root".
Other factoring techniques might involve grouping or using the quadratic formula in scenarios where simple factors aren't apparent. Recognizing common patterns like perfect square forms (which is this case) or sum/difference of squares can speed up factoring greatly.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operation symbols. They form the foundation for equations and are used to represent real-world quantities.
In our factorization exercise, the expression \( x^2 + 12x + 36 \) is a classic example of a quadratic algebraic expression. Interacting with these expressions often involves understanding their structure and finding ways to simplify or manipulate them through different operations.
One useful property is the ability to rewrite an expression in different yet equivalent forms, such as factoring. This can suggest or illuminate the problem's nature, such as revealing roots or simplifying calculations.
Understanding these expressions involves recognizing terms, coefficients, and constants. It's important to be comfortable with the distributive property, which allows an expression's expansion from factored form, and its reverse, used for factoring.