Problem 5 - Precalculus : Building Concepts and Connections

Question

Find the constant term needed to make \(x^{2}-6 x\) a perfect square trinomial.

Step-by-Step Solution

Answer

The constant term needed to make \(x^{2}-6 x\) a perfect square trinomial is 9.

Step 1: Identifying the form and a value

First, compare the given expression \(x^{2}-6 x\) to a perfect square trinomial pattern \(a^{2} - 2ab + b^{2}\). Here, 'a' is obviously 'x', because \(x^{2}\) is the first term of the given polynomial and also first term of the perfect square trinomial pattern.

Step 2: Determining 2ab and b value

In a perfect square trinomial, the middle term, which is -6x in this case, is equal to 2ab. We know a = x and we are to find b. The expression -6x represents -2ab, so we can solve for b by dividing -6x by -2a, which is -2x: \(b= -6x/-2x = 3\). So, the b value needed to form a perfect square is b = 3.

Step 3: Determining \(b^{2}\)

Once b is found in Step 2, we need to square it (calculate \(b^{2}\)) because in a perfect square trinomial the last term is \(b^{2}\). Here, \(b^{2} = (3)^2 = 9\). So, the constant term needed to form a perfect square trinomial is 9.

Key Concepts

Polynomial Expressions

Polynomial expressions are algebraic expressions that contain variables, coefficients, and exponents. These expressions can have one or multiple terms, and each term is composed of:

• A coefficient, which is a constant number.
• Variables, such as x or y, raised to a non-negative integer exponent.

For example, in the expression \(3x^2 - 5x + 2\), there are three terms: \(3x^2\), \(-5x\), and \(2\). The degree of a polynomial is determined by the highest exponent in the expression, which in this example is 2. Therefore, it’s a quadratic polynomial. Understanding the basic structure of polynomials helps in classifying them and in performing operations like addition, subtraction, and multiplication.

Quadratic Expressions

Quadratic expressions are a subset of polynomial expressions where the highest power of the variable is 2. They usually take the standard form \(ax^2 + bx + c\) where:

• \(a\), \(b\), and \(c\) are coefficients.
• \(a eq 0\) to ensure there is a quadratic term.

A key characteristic of quadratic expressions is their ability to represent parabolic graphs when plotted. They can take on different forms such as:

• General form: \(ax^2 + bx + c\)
• Vertex form: \(a(x-h)^2 + k\)
• Factored form: \(a(x-r)(x-s)\)

Each representation provides unique insights into the properties of the quadratic function, like its vertex or roots. Mastering quadratic expressions is crucial as they form the foundation for topics like solving quadratic equations and graphing parabolas.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique is particularly helpful for solving quadratic equations and for deriving the quadratic formula. The process involves:

• Rewriting the quadratic expression in the form of \(ax^2 + bx\).
• Taking the coefficient of the linear term \(b\), dividing it by 2, and then squaring it to find the constant term \(c\).
• Adding and subtracting this constant term \(c\) within the expression to maintain equality.

For instance, with the expression \(x^2 - 6x\), take half of -6, which is -3, and square it to get 9. Hence, adding 9 completes the perfect square trinomial, transforming it into \((x-3)^2 = x^2 - 6x + 9\). This method simplifies many algebraic processes and provides a pathway to solve equations by observing the expression’s simpler square form.