Problem 24 - Algebra 2

Question

Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. \(x^{2}+16 x+c\)

Step-by-Step Solution

Answer

The value of c is 64, making the trinomial \((x+8)^2\).

Step 1: Understand what a perfect square is

A trinomial is a perfect square if it can be written in the form \((ax + b)^2\) which expands to \(a^2x^2 + 2abx + b^2\). Here, we need to identify the value of \(c\) that will make \(x^2 + 16x + c\) fit this format.

Step 2: Identify the middle term

For the trinomial to be a perfect square, \(2ab\) from the expanded form must match the middle term of the trinomial which is 16x here. Therefore, \(2ab = 16x\). Since \(a\) in our trinomial is 1, we equate \(2 \cdot 1 \cdot b\) to 16 and solve for \(b\).

Step 3: Solve for b

Solving \(2b = 16\), we divide both sides by 2 to find \(b\). This gives us \(b = 8\).

Step 4: Find c using the value of b

Knowing that \(b = 8\), recall that a perfect square trinomial \((x + b)^2\) expands to \(x^2 + 2bx + b^2\). Therefore, \(c = b^2 = 8^2 = 64\). So, \(c = 64\).

Step 5: Write the trinomial as a perfect square

With \(c\) found, the trinomial \(x^2 + 16x + 64\) is the same as \((x + 8)^2\) when expanded. Thus, \(x^2 + 16x + c = (x+8)^2\) when \(c = 64\).

Key Concepts

Trinomial

A trinomial is a type of polynomial that consists of three terms, hence the prefix "tri-" which signifies three. In mathematical expressions, trinomials are presented in the form \(ax^2 + bx + c\). This structure contains three important components:

• The term \(ax^2\) is the quadratic term.
• The term \(bx\) is the linear term.
• The term \(c\) is the constant term.

Each part plays a significant role in determining the nature and shape of the graph of the trinomial. Understanding the components of a trinomial is crucial for more complex operations such as factorization or expansion. They act as building blocks in algebra, often used for solving quadratic equations and modelling various physical phenomena.

Square of a Binomial

The square of a binomial is a significant concept in algebra that frequently simplifies problems and helps us understand more complex expressions. A binomial is an expression that consists of two terms. When we square a binomial, it means we are multiplying the binomial by itself. The formula for the square of a binomial \((ax + b)^2\) is:

• It initially transforms into \((ax + b)(ax + b)\).
• When expanded, it is expressed as \(a^2x^2 + 2abx + b^2\).

The result is a perfect square trinomial, just like what we aimed to achieve in our given exercise. For example, squaring the binomial \((x + 8)\) yields the perfect square trinomial \(x^2 + 16x + 64\). This highlights how understanding the square of a binomial can help us identify or construct a perfect square trinomial.

Expanding Trinomials

When we refer to expanding trinomials, we're detailing a process where an expression in factored form is fully multiplied out. While in reverse, forming a trinomial into a square form simplifies handling the expression, expanding a trinomial involves expressing it completely in polynomial form.To expand a trinomial that is a perfect square, you would write it as a product of two binomials, such as:

• Starting from \((x + 8)^2\).
• Expanding it into \((x + 8)(x + 8)\).
• Finally, resulting in \(x^2 + 16x + 64\) once the binomials are multiplied out.

Expanding is crucial in algebra for solving equations, simplifying expressions, or verifying the work related to conversions and transformations of algebraic expressions.