Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-8,\) complete the square of the quadratic expression. $$ x^{2}-12 x $$

Step-by-Step Solution

Answer

The expression completed by the square is \((x - 6)^2 - 36\).

Step 1: Identify the Quadratic Coefficient

First, we need to identify the coefficient of the linear term in the quadratic expression. The expression given is \(x^2 - 12x\). The linear coefficient here is \(-12\).

Step 2: Calculate Half of the Linear Coefficient

Next, we take the linear coefficient, which is \(-12\), divide it by \(2\). This gives us \(-6\).

Step 3: Square the Result

After obtaining \(-6\), we square this result to add a perfect square form to our expression. \((-6)^2 = 36\).

Step 4: Complete the Square

Now, rewrite the expression by adding and subtracting the squared result we computed in the previous step within the expression. So, \(x^2 - 12x\) is rewritten as \(x^2 - 12x + 36 - 36\).

Step 5: Express the Quadratic as a Perfect Square

We can express the expression as a perfect square trinomial. That is, the expression \(x^2 - 12x + 36\) becomes \((x - 6)^2\). This is because \(x^2 - 12x + 36 = (x - 6)(x - 6)\).

Step 6: Write the Completed Expression

Finally, the completed form of the original expression \(x^2 - 12x\) is given by \((x - 6)^2 - 36\).

Key Concepts

Quadratic Expressions

A quadratic expression is an algebraic expression of the form \( ax^2 + bx + c \), where \( a eq 0 \). It represents a polynomial of degree two. In simpler terms, it includes at least one squared variable. Quadratic expressions are foundational in algebra and critical to understanding parabolic graphs. Understanding the basic components of a quadratic expression is important:

• Quadratic Term: The term with \( x^2 \), which determines the parabola's direction: upward if \( a \) is positive, downward if negative.


• Linear Term: The \( bx \) term influences the axis of symmetry and shifting of the parabola.


• Constant Term: The \( c \) term provides the y-intercept of the parabola.

For the expression \( x^2 - 12x \), the absence of a constant term (where \( c = 0 \)) implies it passes through the origin or is vertically shifted if modified with additional terms.

Perfect Square Trinomials

Perfect square trinomials are special forms of quadratic expressions that can be expressed as the square of a binomial. They are fundamental in algebra for simplifying expressions and solving equations. A perfect square trinomial generally looks like: \( (x + n)^2 \) or \( (x - n)^2 \). To identify or form a perfect square trinomial, follow these steps:1. Identify the linear component \( b \). In our example, it is \(-12\).2. Find half of \( b \), which is \(-6\) in the example, then square it to get \( 36 \).3. Add and subtract this square from the quadratic expression.4. Rewrite it in the form \( (x + n)^2 \) or \( (x - n)^2 \).In \( x^2 - 12x \), by adding \( 36 \), we form \( x^2 - 12x + 36 \), transforming it into the perfect square trinomial \( (x - 6)^2 \). This makes it easier to handle for further algebraic operations.

Factoring

Factoring is the process of breaking down complex expressions into simpler multiplicative components, which are easier to solve or compute with. For quadratic expressions, factoring transforms them into products of binomials. This is beneficial for finding roots or zeros of the quadratic equation. To factor a perfect square trinomial, you need to:

• Recognize the trinomial as a perfect square, like \( x^2 - 12x + 36 \).
• Break it into a repeated binomial: \( (x - 6)(x - 6) \), or simply \( (x - 6)^2 \).


• Understanding its factored form provides insights into the squared nature of the expression and its graph.

The factored expression \( (x - 6)^2 \) directly shows that the expression has a double root at \( x = 6 \). This means that at \( x = 6 \), the parabola touches the x-axis, emphasizing its symmetry and vertex point.