Problem 36 - Algebra 1: Concepts and Skills

Question

PERFECT SQUARES Factor the expression. $$ 36 m^{2}-84 m+49 $$

Step-by-Step Solution

Answer

\((6m - 7)^{2}\)

Step 1: Identify the Terms

In the equation \(36 m^{2}-84 m+49\), \(36 m^{2}\) is the first term, \(-84 m\) is the second term and \(49\) is the third term. Let us confirm if this is a perfect square trinomial.

Step 2: Calculate the Roots

For a perfect square trinomial, the square root of the first term should be 'a' and the square root of the last term should be 'b'. In this case, the square root of the first term \(36 m^{2}\) is \(6m\), and the square root of the third term \(49\) is \(7\). Let's try to express the middle term as a product of \(2ab\).

Step 3: Confirm Middle Term

Calculate \(2ab\): \(2(6m)(7)\), which equals \(-84m\). This does indeed match the middle term of our trinomial, confirming it's a perfect square trinomial.

Step 4: Factorize the Trinomial

The quadratic trinomial is now confirmed as a perfect square. According to the rule of factorization of perfect squares, it can be rewritten as \((a - b)^{2}\). So, \(36 m^{2}-84 m+49\) can be factored into \((6m - 7)^{2}\)

Key Concepts

Algebraic Expressions

In algebra, an algebraic expression is a mathematical phrase that can include numbers, variables (like 'm' in our example), and operation symbols. It's essential to understand that an expression represents a value that can change depending on the variables involved. For instance, the expression given in the exercise, \(36 m^{2}-84 m+49\), consists of terms combined by subtraction operations.

Each term is a product of a coefficient (a numerical factor) and a variable raised to a power, as in \(36 m^{2}\), where 36 is the coefficient and \(m^{2}\) indicates that 'm' is squared. Recognizing the structure of algebraic expressions is pivotal as it sets the foundation for various operations, including factorization, which is central to solving equations and simplifying expressions in algebra.

Quadratic Trinomial Factorization

Quadratic trinomial factorization is a process where a trinomial of the second degree — a polynomial with three terms, the highest degree of which is two — is expressed as the product of two binomials. Factorization is beneficial as it breaks down more complex expressions into products of simpler ones, making further calculations or solving equations manageable.

For the trinomial \(36 m^{2}-84 m+49\), which is quadratic (its highest degree is 2), the factorization process involves finding two binomials that multiply to give the original trinomial. This exercise employs a specific technique for factorization because the given quadratic trinomial is a perfect square, meaning it derives from squaring a binomial. Recognizing a perfect square trinomial is key to simplifying it into a compact, squared binomial form—like going from \(a^{2} - 2ab + b^{2}\) back to \((a - b)^{2}\).

Perfect Squares

The concept of perfect squares is foundational in algebra. A perfect square is the product of a number or a variable with itself, such as \(49=7^{2}\) or \(m^{2}=m\times m\). In our exercise, the trinomial \(36 m^{2}-84 m+49\) is considered a perfect square trinomial because it fits the form \(a^{2} - 2ab + b^{2}\), where \(a\) and \(b\) are perfect squares themselves.

Understanding perfect squares not only aids in factorization but also in simplifying square roots and solving quadratic equations. To factor a perfect square trinomial, one should recognize the structure of the squared terms and the middle term, which is crucial. For instance, if \(a = 6m\) and \(b = 7\), then the middle term should be \(-2ab = -84m\), confirming our expression can be factorized neatly into \((6m - 7)^{2}\), a much simpler representation that can be used to solve algebraic equations or inequalities.