Problem 47 - Algebra 2

Question

Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. \(x^{2}-2.4 x+c\)

Step-by-Step Solution

Answer

The value of \(c\) is 1.44, and the trinomial is \((x - 1.2)^2\).

Step 1: Identify the formula for a perfect square trinomial

A perfect square trinomial is of the form \((a-b)^2 = a^2 - 2ab + b^2\). Our goal is to rewrite \(x^2 - 2.4x + c\) in this form.

Step 2: Match the trinomial to the perfect square formula

The given trinomial is \(x^2 - 2.4x + c\). In the formula \((a-b)^2 = a^2 - 2ab + b^2\), compare the term \(2ab\) with \(2.4x\): \(2b = 2.4\).

Step 3: Solve for b

From \(2b = 2.4\), solve for \(b\) by dividing both sides by 2 to get \(b = 1.2\).

Step 4: Calculate \(b^2\) to find c

Now that we know \(b = 1.2\), calculate \(b^2 = (1.2)^2 = 1.44\). Thus, \(c = b^2 = 1.44\).

Step 5: Write the trinomial as a perfect square

Substitute \(c = 1.44\) back into the trinomial: \(x^2 - 2.4x + 1.44\). This can be rewritten as \((x - 1.2)^2\).

Key Concepts

Trinomial

A trinomial is an algebraic expression composed of three terms. In algebra, you often encounter trinomials in the format of \(ax^2 + bx + c\). Each term is connected by addition or subtraction. A special case of trinomials is the perfect square trinomial, which can be factored into a binomial squared.

• The concept revolves around understanding how to structure an expression that can simplify into the square of another binomial expression.
• In the original exercise, the trinomial is \(x^2 - 2.4x + c\).
• The aim is to adjust this trinomial so that it reflects the structure of a perfect square trinomial, allowing it to be simplified into a binomially squared form.

Trinomials are fundamental in factoring and simplifying algebraic expressions. Mastery of these concepts not only simplifies solving equations but also enhances understanding of the relation between polynomials and their factorized forms.

Binomial

A binomial is a simple algebraic expression that contains exactly two terms, such as \(a + b\) or \(m^2 - 5n\). Binomials are the building blocks of more complex expressions like trinomials.

• These expressions are crucial in algebra because they can hold the key to simplifying larger equations.
• When a trinomial is identified as a perfect square, it can be represented as a squared binomial.
• In the given problem, the trinomial \(x^2 - 2.4x + 1.44\) is actually \((x - 1.2)^2\), which is a binomial squared.

Understanding how to convert a trinomial into a binomial square allows one to easily manage complex algebraic expressions and solve quadratic equations efficiently.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They are the foundation of algebra and are used to represent real-world problems in a mathematical format.

• An algebraic expression like \(x^2 - 2.4x + c\) helps us understand relationships between numbers and variables.
• They are formed by combining constants, variables, and algebraic operations like addition, subtraction, multiplication, and division.
• In the exercise, the goal was to manipulate the expression so it becomes a perfect square, which involves understanding how to rearrange and complete it to fit the desired form.

Algebraic expressions allow us to simplify, evaluate, and solve equations. Proficiency in working with these expressions expands one's ability to handle more challenging problems, both in academic settings and real-life scenarios.