Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-8,\) complete the square of the quadratic expression. $$ x^{2}+6 x $$

Step-by-Step Solution

Answer

The completed square form is \((x + 3)^2 - 9\).

Step 1: Identify the Coefficient

In the given quadratic expression \(x^2 + 6x\), the coefficient of \(x\) is \(6\). This is the term that you need to focus on when completing the square.

Step 2: Halve the Coefficient

Take the coefficient of \(x\), which is \(6\), divide it by \(2\) to get \(3\). This means the expression will revolve around \((x + 3)^2\) when completed.

Step 3: Square the Result

Square the result from Step 2. In this case, \(3^2 = 9\). This value will be both added and subtracted to complete the square.

Step 4: Add and Subtract the Square

Write the expression as \(x^2 + 6x + 9 - 9\). By doing this, you balance the expression while setting it up for factoring.

Step 5: Factor the Perfect Square

Now, express \(x^2 + 6x + 9\) as a perfect square. This can be written as \((x + 3)^2\). So the expression becomes \((x + 3)^2 - 9\).

Key Concepts

Quadratic Expression

Quadratic expressions form a cornerstone in algebra. They generally appear in the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents a variable. Understanding how these expressions work is vital to mastering high school algebraic concepts.
Breaking down a quadratic expression means taking note of these three components:

• The quadratic term: This is the term containing \(x^2\) and is where the 'quadratic' name comes from.
• The linear term: This part has the single \(x\).
• The constant term: A standalone number with no \(x\) involved at all.

The quadratic expression seen in the original exercise is \(x^2 + 6x\). Here, the quadratic term is \(x^2\), the linear term is \(6x\), and interestingly, the constant term is absent. Understanding this initial layout helps us transition effectively into more advanced steps, such as completing the square.

Perfect Square

A perfect square is a significant concept when dealing with quadratic expressions. It originates from the idea of representing numbers as the square of another number. For example, \(9\) is a perfect square because it is \(3^2\).
In algebra, forming a perfect square helps simplify quadratic expressions and solve equations more efficiently.
Completing a square typically involves converting a quadratic expression into one that represents a square of a binomial. That is, an expression in the form \((x + d)^2\). Steps to accomplish this typically include:

• Identifying the middle term coefficient, in this case, \(6\).
• Dividing it by \(2\) and squaring the result, leading to the creation of our perfect square number. For \(6\), this results in \(9\).

Once this extra term is added and subtracted appropriately, the quadratic expression can then be rewritten as a binomial square like \((x + 3)^2\), effectively simplifying it to a perfect square.

Factoring

Factoring is an indispensable tool in algebra used to simplify expressions or solve equations. When we talk about factoring quadratic expressions, we essentially mean breaking down an expression into products of simpler ones. This method allows us to easily work through these expressions and make them more manageable.
For any quadratic like \(x^2 + 6x + 9\), we can use factoring by rewriting it into the product of binomials. By seeing the expression as \((x + 3)^2\), it has been factored from the expanded form.
Factoring is a skill developed through practice. It's essential to recognize patterns such as perfect squares or other binomial products. Once an expression is factored, it can often reveal solutions to equations much more readily or help in further simplifications.

Algebra

Algebra is a broad field of mathematics concerning symbols and the rules for manipulating these symbols to solve equations or represent real-world scenarios. At its heart, algebra enables the formulation and solving of equations.
Quadratic expressions, like \(x^2 + 6x\), are just one of many types of algebraic equations students will encounter. They appear in both everyday problems and more complex mathematical explorations.
In essence, algebra is a toolkit:

• It provides expressions, such as quadratics, through which real-world issues can be examined and discussed.
• It equips learners with methods for rearranging equations, such as completing the square, to learn more about what the equations describe or solve for specific variables.

By mastering the rules and concepts of algebra, students can handle a wide range of mathematical problems efficiently, confidently transferring knowledge across different areas.