Problem 8 - Algebra 2

Question

Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. \(x^{2}-12 x+c\)

Step-by-Step Solution

Answer

The value of c is 36, and the trinomial can be expressed as \((x - 6)^2\).

Step 1: Understand what a perfect square trinomial is

A perfect square trinomial can be expressed in the form \((a - b)^2 = a^2 - 2ab + b^2\). For the given trinomial to be a perfect square, it should match this structure.

Step 2: Identify components of the given trinomial

The given trinomial is \(x^2 - 12x + c\). Here, the term \(x^2\) corresponds to \((x)^2\), and the linear term \(-12x\) corresponds to \(-2bx\). We need to determine the value of \(b\).

Step 3: Solve for b using the linear term

The linear term is \(-12x\), which corresponds to \(-2bx\). So, \(-2b = -12\). By solving for \(b\), we divide both sides by \(-2\) getting \(b = 6\).

Step 4: Find the value of c

The value of \(c\) is given by \(b^2\). Since \(b = 6\), then \(b^2 = 6^2 = 36\). Thus, \(c = 36\).

Step 5: Write the trinomial as a perfect square

With \(c = 36\), the trinomial \(x^2 - 12x + c = x^2 - 12x + 36\) becomes \((x - 6)^2\). Therefore, \(x^2 - 12x + 36\) can be written as \((x - 6)^2\).

Key Concepts

Trinomial Structure

To understand perfect square trinomials, you first need to get familiar with the trinomial structure. A trinomial is a polynomial with three terms, typically written as

• the first term: the square of a variable, like \(x^2\)
• the second term: a linear term, such as \(-12x\)
• the third term: a constant, represented here as \(c\).

A perfect square trinomial follows the specific pattern \((a-b)^2 = a^2 - 2ab + b^2\). It's essential because it helps easily factor the polynomial later. By identifying each part of the trinomial, you can see how to rearrange or complete it to achieve a perfect square formation.

Factoring Trinomials

Factoring trinomials is a method used to break down quadratic expressions into simpler factors. When you have a perfect square trinomial, such as

• \(x^2 - 12x + 36\), which results in:

The expression can be written as \((x - 6)^2\). This means the trinomial can be factored into two identical binomial expressions, because it represents the square of a binomial. Factoring becomes easy when you identify the structure of a perfect square, as it matches the form \((a - b)^2 = a^2 - 2ab + b^2\). When working with general trinomials, recognizing this structure can help simplify and solve quadratic equations with ease.

Completing the Square

Completing the square is a technique used to transform a quadratic equation into a perfect square trinomial. This process is immensely helpful when dealing with quadratic expressions. Here's how it works:

• Start with a quadratic expression like \(x^2 - 12x + c\).
• Identify the linear term, which is \(-12x\), and set it equal to \(-2bx\).

To complete the square, solve for \(b\). In this case, solve \(-2b = -12\), which gives \(b = 6\). The next step is to add \(b^2\) to complete the trinomial, so \(c = b^2 = 36\). Adding this \(c\) value ensures the expression forms a perfect square trinomial, leading to \((x - 6)^2\). This method not only simplifies factoring but also helps in solving quadratic equations.

Quadratic Expressions

Quadratic expressions are mathematical statements that include a variable squared. They typically take the form \(ax^2 + bx + c\). This form is vital for understanding and solving complex math problems.

• The quadratic term, such as \(x^2\), establishes the degree of the expression as 2.
• The linear term, \(bx\), contributes to creating the symmetry required for factoring.
• The constant term, \(c\), is crucial in forming perfect square trinomials.

In our example, \(x^2 - 12x + c\) transforms into a solvable equation by turning it into a perfect square \((x - 6)^2\). Mastering quadratic expressions empowers you to tackle a wide array of math problems by leveraging their structural properties.