Problem 8 - Introductory Algebra for College Students

Question

Complete the square for binomial. Then factor the resulting perfect square trinomial. \(x^{2}-x\)

Step-by-Step Solution

Answer

The factored form of the given expression \(x^{2}-x\) is \(x*(x-1)\)

Step 1: Add and Subtract a New Term to Complete the Square

In the binomial \(x^{2}-x\), the second term is -x. Half of -1 (the coefficient of x) is -0.5 and its square is \(0.25\). So, add and subtract \(0.25\) inside the square function. It should look like this: \(x^{2}-x+0.25-0.25\) which simplifies to \( (x^{2}-x+0.25)-0.25\)

Step 2: Simplify the Expression to a Perfect Square Trinomial

Now, we can write the first part of the equation as the square of a binomial. Because \((x - 0.5)^2= x^{2}-x+0.25\), so the complete square of this binomial is \((x - 0.5)^{2}-0.25\)

Step 3: Factor the Perfect Square Trinomial

Since we have a square in the equation, we can factor the trinomial. The factored form of this perfect square trinomial is \((x -0.5 +0.5)(x -0.5 -0.5) = (x)(x-1)\)

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special form of a trinomial that can be expressed as

• The square of a binomial, like \[ (a+b)^2 = a^2 + 2ab + b^2 \]
• Or \[ (a-b)^2 = a^2 - 2ab + b^2 \]

In the exercise, the trinomial derived in Step 2 is \[ x^2 - x + 0.25 \], which can be rewritten as a perfect square trinomial, \[ (x - 0.5)^2 \].
To identify a perfect square trinomial, one must:

• Ensure the first and last terms are perfect squares.
• Check if the middle term is twice the product of the square roots of the first and last terms.

This method helps convert a simple quadratic into a form that can be easily factored or used in solving equations.

Factoring Binomials

Factoring binomials involves expressing a binomial as a product of its factors. In our exercise, after completing the square and forming the perfect square trinomial \[ (x - 0.5)^2 \], we factor it further. In the end, this expression simplifies via factoring to \[ (x)(x-1) \].
The steps for factoring binomials generally include:

• Identifying common factors if any exist.
• Recognizing patterns like differences of squares, sum/difference of cubes, etc.
• Applying the distributive property, also known as factoring by grouping.

For quadratics named as binomials, such as in this case, it often results in recognizing the expression as a factorable perfect square or a recognizable pattern.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They do not have equal signs, unlike equations. An example from the exercise would be \[ x^2 - x \].
The main parts of algebraic expressions include:

• Terms: Single mathematical elements separated by plus or minus signs.
• Coefficients: Numbers multiplying the variables in the expressions.
• Variables: Symbols representing numbers, often 'x', 'y', etc.

Algebraic expressions can be manipulated in various ways through operations such as addition, subtraction, and factoring, which are crucial for simplifying expressions like those in the exercise.
Understanding these manipulations, such as completing the square, offers a refined approach to solving and analyzing these expressions.