Problem 24 - Elementary and Intermediate Algebra

Question

Factor. $$ m^{2}+12 m+36 $$

Step-by-Step Solution

Answer

The expression factors to \( (m+6)^2 \).

Step 1: Identify the Form

The expression to be factored is a quadratic trinomial. It has the form \( ax^2 + bx + c \), where \( a = 1 \), \( b = 12 \), and \( c = 36 \).

Step 2: Determine if it is a Perfect Square Trinomial

Check if the expression is a perfect square trinomial. A trinomial \( a^2 + 2ab + b^2 \) can be rewritten as \( (a+b)^2 \). Here, \( m^2 \) is \( (m)^2 \) and \( 36 \) is \( (6)^2 \). The middle term needs to be checked: \( 12m = 2 imes m imes 6 \), which satisfies the form \( 2ab \). Thus, it is a perfect square trinomial.

Step 3: Rewrite the Expression as a Square

Since the trinomial is a perfect square, it can be written in the form \( (a+b)^2 \). Thus, \( m^2 + 12m + 36 \) can be rewritten as \( (m+6)^2 \).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that can be turned into a squared binomial. This means that the expression can be expressed as the square of a binomial.
To identify whether a trinomial is a perfect square trinomial, you need to focus on its specific properties. The standard form of a quadratic trinomial is:

• \( ax^2 + bx + c \)
• For it to be a perfect square trinomial, it must fit the pattern:\( a^2 + 2ab + b^2 \)

This pattern allows us to factor the trinomial into \((a + b)^2\).
For example, if we have a quadratic expression like \( m^2 + 12m + 36 \), we check each term:

• The first term \( m^2 \) is \((m)^2\),
• The last term \( 36 \) is \((6)^2\),
• The middle term \( 12m \) equals \( 2 \times m \times 6 \).

These components perfectly match the pattern \( a^2 + 2ab + b^2 \), confirming it as a perfect square trinomial.

Factoring

Factoring is a key technique used to simplify quadratic expressions. Essentially, it involves rewriting the expression as a product of simpler expressions.
This process is especially useful in solving equations, as it allows us to set each factor to zero and solve for the variable.
To factor a quadratic trinomial like \( m^2 + 12m + 36 \), one effective way is to first check if it is a perfect square trinomial. If it is, factoring becomes a straightforward task:

• You write the trinomial using the form \((a+b)^2\), where here \(m^2 = (m)^2 \) and \( 36 = (6)^2 \).
• Use the middle term \( 12m = 2 \times m \times 6\), supporting it as \( 2ab \).

Once confirmed, the factoring of \( m^2 + 12m + 36 \) becomes:

• Simply \((m + 6)^2\).

By factoring, solving or simplifying expressions becomes an easier process.

Quadratic Expressions

Quadratic expressions are mathematical expressions where the highest exponent of the variable is two. They take the general form \( ax^2 + bx + c \).
These expressions are common in algebra and have many applications, from physics to finance.
The process of working with quadratic expressions often involves factoring them into simpler, easily solvable forms.

• This can involve recognizing perfect squares, implementing the quadratic formula, or completing the square.
• Each technique provides a method to rewrite or solve the quadratic expression.

For example, in the expression \( m^2 + 12m + 36 \), we apply our knowledge of perfect square trinomials to simplify it effectively into \((m + 6)^2\).
All quadratic expressions can generally be resolved or simplified with systematic approaches that explore their structures.