Problem 27 - Algebra 2

Question

Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. \(x^{2}+7 x+c\)

Step-by-Step Solution

Answer

The value of \(c\) is \(\frac{49}{4}\), and the trinomial is \((x + \frac{7}{2})^2\).

Step 1: Understanding the Perfect Square Trinomial

A perfect square trinomial is of the form \(a^2 + 2ab + b^2\), which can be rewritten as \((a + b)^2\). We need to adjust \(x^2 + 7x + c\) into this structure.

Step 2: Identify 'a' and '2ab' Values

In the trinomial \(x^2 + 7x + c\), identify \(a\) and \(2ab\):- Here, \(a = x\).- Therefore, the term \(7x\) should represent \(2ab\). Thus, \(2ab = 7x\).

Step 3: Solving for 'b'

Since \(2ab = 7x\) and \(a = x\), we can rearrange it to find \(b\):\(2x \cdot b = 7x\),Cancel \(x\) from both sides: \(2b = 7\).Solve for \(b\): \(b = \frac{7}{2}\).

Step 4: Find 'c' to Complete the Perfect Square

The term \(c\) in the trinomial must be \(b^2\) to complete the perfect square:Calculate \(c = b^2 = \left(\frac{7}{2}\right)^2 = \frac{49}{4}\).

Step 5: Write the Trinomial as a Perfect Square

Now, substitute \(c\) back into the trinomial and express it as a perfect square:\(x^2 + 7x + \frac{49}{4} = \left(x + \frac{7}{2}\right)^2\).

Key Concepts

Trinomial

A trinomial is an algebraic expression composed of three terms. These terms are typically combined using addition or subtraction. In the context of algebra, the structure of a standard trinomial looks like this:
\[ ax^2 + bx + c \]
In this format, each variable and coefficient plays a vital role in algebraic operations. Let's break it down:

• '\( a \)' is the coefficient of the quadratic term \( x^2 \).
• '\( b \)' represents the coefficient of the linear term \( x \).
• '\( c \)' is the constant term.

These components can change depending on the specific problem you are tackling, but the overall goal is to manipulate these terms effectively. In our given exercise, \( x^2 + 7x + c \), identifying these terms correctly is key to solving the problem. By recognizing the patterns within a trinomial, you can find solutions such as identifying perfect squares, which greatly simplifies solving quadratic equations.

Algebra

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is a key area that impacts various math topics, including solving equations and understanding expressions. Let's delve into some critical aspects:

• Simplifying expressions: Algebra allows you to reduce complex expressions to simpler forms. This is done by following specific rules and properties, such as the distributive property or combining like terms.
• Solving equations: One of the fundamental purposes of algebra is to find the value of unknowns represented by variables. In our exercise, solving for \( c \) involves applying algebraic rules to achieve the desired result of a perfect square trinomial.
• Manipulating polynomials: Polynomials are algebraic expressions made up of terms combined through addition or subtraction. Mastery of working with polynomials allows one to handle tasks such as factoring, expanding, and solving for roots.

In the exercise provided, algebra helps us apply transformations to the trinomial, turning it into a perfect square trinomial. By understanding the rules, we can solve problems efficiently and effectively.

Quadratic Equations

Quadratic equations are a specific type of polynomial equation of the second degree. They take the standard form:
\[ ax^2 + bx + c = 0 \]
These equations are central to algebra and offer a range of solutions, including direct calculation methods, such as the quadratic formula or graphical interpretations.Key features of quadratic equations include:

• The term \( ax^2 \) represents the quadratic component and determines the 'parabolic' shape of its graph.
• The coefficients \( a, b, \) and \( c \) affect the position and orientation of the parabola.
• Roots of the equation, which are the solutions of the equation where the graph intersects the x-axis.

For the exercise \(x^2 + 7x + c\), solving it as a perfect square is advantageous. This process involves recognizing it as a trinomial and effectively transforming it. By arranging it in a perfect square form, namely \((x + b)^2\), you simplify the problem, making it easy to solve or graph. Mastery of these concepts allows you to tackle complex algebraic problems with confidence and ease.