Problem 38 - Algebra 2

Question

Factor each expression. $$ t^{2}-14 t+49 $$

Step-by-Step Solution

Answer

\((t - 7)^2\)

Step 1: Identify a perfect square trinomial

Note that the given quadratic expression is a trinomial which resembles the form of a perfect square trinomial \(a^2 - 2ab + b^2\), where \(a\) is the square root of the first term and \(b\) is the square root of the last term.

Step 2: Find the square roots of the first and last terms

The first term is \(t^2\), whose square root is \(t\). The last term is 49, whose square root is 7. Therefore, \(a = t\) and \(b = 7\).

Step 3: Write the factored form

Since the middle term is negative \(2ab\), and both \(a\) and \(b\) are positive, it means that the trinomial is a perfect square of the binomial \(a - b\). Thus, the expression can be factored as \( (t - 7)^2\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (addition, subtraction, multiplication, and division). They are foundational in algebra and help us describe relationships and changes. For example, the expression t^2 - 14t + 49 involves the variable t, which stands for a number that can change, or 'vary'.

When we start to decipher an algebraic expression, we focus on its components, such as:

• Constants: Non-changing numbers, like the 49.
• Variables: Symbols that represent numbers, like t in our example.
• Coefficients: Numbers that multiply the variables, such as the 14 in front of t.
• Exponents: Indicating how many times a variable is multiplied by itself, as seen in t^2.

Identifying these parts helps us understand the structure of the expression and prepares us for operations like factoring.

Quadratic Expressions and Their Characteristics

Quadratic expressions are a type of algebraic expression that includes a variable raised to the second power (squared) as its highest power. The general form of a quadratic expression is ax^2 + bx + c, with a, b, and c being constants.

Quadratic expressions often represent parabolas when graphed on a coordinate plane. One of the key tasks in studying quadratics is factoring. This process turns the quadratic expression into a product of binomials or simpler expressions. Factoring can simplify the expression and is crucial for solving quadratic equations.

For the quadratic expression t^2 - 14t + 49, we can see it has the quadratic term t^2, the linear term -14t, and a constant term 49. The coefficients here are 1 for t^2, -14 for t, and 49 is the constant.

Perfect Square Trinomial Factorization

Perfect square trinomial factorization is a specific type of factoring where the expression resembles the form a^2 ± 2ab + b^2, which factors into (a ± b)^2. This occurs when the trinomial is the square of a binomial.

In the provided example t^2 - 14t + 49, we first identify if it's a perfect square trinomial by checking if the first and last terms are perfect squares and if the middle term is twice the product of their square roots.

Here's a step-by-step approach:

• First term: t^2 is a perfect square of t.
• Last term: 49 is a perfect square of 7.
• Middle term: -14t is twice the product of t and 7 (as -2 * t * 7 gives -14t).

With these conditions met, the expression can indeed be factored as (t - 7)^2, which is the binomial t - 7 multiplied by itself.