Problem 27 - Algebra 1: Concepts and Skills

Question

Find the term that should be added to the expression to create a perfect square trinomial. $$ x^{2}-6 x $$

Step-by-Step Solution

Answer

The term that should be added to the expression to create a perfect square trinomial is \(9\). The complete perfect square trinomial is \(x^2 - 6x + 9\).

Step 1: Identify the form of the perfect square trinomial

First, need to identify the form of the perfect square trinomial which can be written as \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). The given expression is \(x^2 - 6x\). It seems to match with the latter form. Here, \(a = x\) and the coefficient of \(x\) is -6 which actually corresponds to \(2ab\). This means that \(2ab = -6x\) where \(b\) is the constant that should be added.

Step 2: Calculate for b

Next, solve for \(b\) from the equation \(2ab = -6x\). Since \(a = x\), the equation becomes \(2xb = -6x\). Simplifying this gives \(b = -6 / 2 = -3\).

Step 3: Find the square of b

To get a perfect square trinomial, need to add the square of \(b\) as our constant term to the existing binomial. Calculate the square of \(b\) using the expression \(b^2\). Here, \(b = -3\). So, \((-3)^2 = 9\).

Key Concepts

Completing the Square

Completing the square is a technique used in algebra to transform a quadratic expression into a perfect square trinomial. This method is especially useful when solving quadratic equations, as it allows you to easily factor the expression or find the roots of the equation.

To complete the square, the key idea is to add (or subtract) a term that turns the expression into a perfect square trinomial. Consider the quadratic expression in the given exercise: \(x^2 - 6x\). We want to manipulate this expression into the form \((x + b)^2\).

Here's a straightforward way to complete the square:

• Identify the coefficient of the linear term (here, it's \(-6\)).
• Divide this coefficient by \(2\), then square the result to get the term that completes the square.
• Add this squared term to the expression to form a perfect square trinomial.

In our example, the term \(9\) completes the square, converting \(x^2 - 6x\) into \((x - 3)^2\). This method not only makes it easier to handle quadratics but also prepares you to leverage other algebraic techniques efficiently.

Quadratic Expression

A quadratic expression is a polynomial of degree two. It typically takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The expression \(x^2 - 6x\) from the exercise is a simple form of a quadratic expression where \(a = 1\) and \(c = 0\).

Quadratic expressions have distinctive characteristics that make them a separate category of polynomials. They typically feature a parabolic graph when plotted, which can open upwards or downwards, depending on the sign of \(a\).

In the context of creating a perfect square trinomial, you focus on transforming the initial quadratic expression by introducing a new term. This converts the expression into an equivalent form that allows you to express it as the square of a binomial.

Recognizing the structure of a quadratic expression is vital in manipulating it effectively. This understanding will also aid in solving quadratic equations, whether using factoring, completing the square, or quadratic formula methods.

Algebraic Manipulation

Algebraic manipulation involves strategically rearranging, factoring, or otherwise transforming an expression or equation to achieve a desired form. When dealing with quadratic expressions, several manipulation techniques can simplify the problem-solving process.

In our example, we transform the expression \(x^2 - 6x\) into a perfect square trinomial by algebraically adding the term \(9\). This addition is carefully calculated to match the pattern \((x - 3)^2\).

The steps include:

• Identifying the current form of the quadratic and what alteration is needed to achieve the perfect square form.
• Solving for \(b\) if necessary (from \(2ab = -6x\) in our exercise, leading us to \(b = -3\)).
• Adding \(b^2\) to complete the square effectively.

Such manipulation is a powerful skill that enables you to simplify expressions, solve equations, and handle more complex algebraic problems with ease and confidence. Through practice, you will become adept at recognizing patterns and selecting appropriate manipulation techniques for different algebraic challenges.