Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-8,\) complete the square of the quadratic expression. $$ x^{2}-3 x $$

Step-by-Step Solution

Answer

The completed square form of the expression is \((x - \frac{3}{2})^2 - \frac{9}{4}\).

Step 1: Identify the Quadratic and Linear Terms

In the expression, \(x^2 - 3x\), identify the quadratic term \(x^2\) and the linear term \(-3x\), ignoring the constant term for now.

Step 2: Determine the Coefficient for Completing the Square

Take the coefficient of the linear term, which is \(-3\), divide it by 2, and square the result. This gives us \((-3/2)^2 = 9/4\).

Step 3: Write the Completed Square Expression

Add and subtract \(9/4\) inside the expression to balance it: \(x^2 - 3x + \frac{9}{4} - \frac{9}{4}\).

Step 4: Factor the Perfect Square Trinomial

The expression \(x^2 - 3x + \frac{9}{4}\) is a perfect square trinomial. It can be factored into \((x - \frac{3}{2})^2\).

Step 5: Combine and Simplify the Expression

Include the subtracted term \(-\frac{9}{4}\) from Step 3 to complete the expression: \((x - \frac{3}{2})^2 - \frac{9}{4}\).

Key Concepts

Quadratic Expression

A quadratic expression is an algebraic expression where the highest exponent of the variable is 2. It usually appears in the form \(ax^2 + bx + c\). In this expression:

• \(a\) is the coefficient of the squared term, \(x^2\).
• \(b\) is the coefficient of the linear term, \(x\).
• \(c\) is the constant term.

For example, in the given exercise, the quadratic expression is \(x^2 - 3x\). Notice that there is no constant term in this example, which makes the process a bit simpler.
Working with quadratic expressions is a fundamental skill in algebra because it allows you to solve quadratic equations, understand functions, and model real-world scenarios. Understanding how to manipulate them helps in various fields and levels of mathematics.

Perfect Square Trinomial

A perfect square trinomial is a special form of a quadratic expression. It can be expressed as the square of a binomial. The general form is: \((ax + b)^2 = a^2x^2 + 2abx + b^2\).When completing the square, we try to make the quadratic expression into a perfect square trinomial. Let's see how:

• Find half of the coefficient of the linear term \(b\).
• Square it to get \(b^2\), which allows you to balance the equation.

In our example, the linear coefficient is \(-3\). First, divide \(-3\) by 2 to get \(-\frac{3}{2}\). Now square it: \(-\frac{3}{2}\)^2 gives you \(\frac{9}{4}\).
Adding and subtracting \(\frac{9}{4}\) within the expression allows us to write it as a perfect square trinomial, which can then be factored into a square of a binomial.

Factoring Quadratics

Factoring quadratics, especially perfect square trinomials, is a vital algebraic skill. It involves rewriting the quadratic expression as the product of two binomials. In many cases, this makes solving, simplifying, or finding the roots of the equation easier.
When we talk about a perfect square trinomial such as \(x^2 - 3x + \frac{9}{4}\), it's already prepared to be factored nicely. This trinomial factors specifically into a square of a binomial:

• \(x^2 - 3x + \frac{9}{4} = (x - \frac{3}{2})^2\).

This is immensely helpful because squared terms are more straightforward to work with in further algebraic operations, such as solving equations or graphing.
In general, practicing the method of completing the square ensures you are always able to transform any quadratic expression into a solvable form, whether it’s initially presented as a perfect square trinomial or not.