Problem 47 - Elementary and Intermediate Algebra

Question

Simplify each expression. Assume that all variables are unrestricted and use absolute value symbols when necessary. See Example 2. $$ \sqrt{y^{2}-2 y+1} $$

Step-by-Step Solution

Answer

\(\sqrt{y^2 - 2y + 1} = |y-1|\)

Step 1: Recognize the Expression Type

The expression inside the square root \( y^2 - 2y + 1 \) is a quadratic expression. We suspect it might be a perfect square trinomial since it has a form that resembles \((a-b)^2 = a^2 - 2ab + b^2\).

Step 2: Identify Perfect Square

Compare \( y^2 - 2y + 1 \) to the form \((a-b)^2\) to see if it matches. Let's rewrite the expression:\[ (y - 1)^2 = y^2 - 2y + 1 \].

Step 3: Simplify the Expression

Now that we know \( y^2 - 2y + 1 = (y-1)^2 \), we can simplify the square root: \( \sqrt{y^2 - 2y + 1} = \sqrt{(y-1)^2} \).

Step 4: Apply the Absolute Value Property

When taking the square root of a squared term \( \sqrt{(y-1)^2} \), it yields \(|y-1|\). This is due to the square root property, which accounts for both positive and negative values of the expression \(y-1\).

Step 5: Write the Final Simplified Expression

Thus, the simplified form of \( \sqrt{y^2 - 2y + 1} \) is \(|y-1|\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be expressed as the square of a binomial.
To identify if a trinomial is perfect, we look for expressions of the form \( a^2 - 2ab + b^2 \) or \( a^2 + 2ab + b^2 \). These correspond to \((a-b)^2\) and \((a+b)^2\) respectively.

Recognizing a perfect square trinomial means you can rewrite it as a binomial squared:

• Identify \(a^2\) and \(b^2\) as perfect square terms.
• Ensure the middle term is \(-2ab\) or \(+2ab\).

Finding a perfect square helps in simplification processes like factoring and solving equations. It allows us to reduce expressions under a square root to more manageable forms.

Square Root Simplification

Simplifying a square root involves reducing the expression inside the radical to its simplest form.
This can free us from complex terms and makes solving equations easier. The square root simplification process can be guided by identifying expressions that are perfect squares.

Steps for square root simplification include:

• Identify the expression under the square root. If it is a perfect square, such as \((a-b)^2\), it can be simplified directly.
• Take the square root of the expression: \( \sqrt{(a-b)^2} = |a-b| \).
• This simplification can expose underlying mathematical relationships and support further computation.

For any squared term inside the radical, remember that its square root is about unraveling that squaring, keeping in mind that negative and positive values should be accounted for.

Absolute Value Property

The absolute value property comes into play when dealing with the square root of a squared expression.
This concept ensures that resulting expressions accurately represent both possible values, positive and negative, of the original expression.

Why use the absolute value?

• Guarantees Non-Negativity: The absolute value of a number represents its distance from zero on a number line, ensuring it is always positive or zero.
• Preserves Original Significance: If \( c = |x| \), \( c \) understands both \( x \) and \( -x \) are valid roots.

When simplifying \( \sqrt{(a-b)^2} \), we use \(|a-b|\) to maintain the correctness of the original equation's possible negative or positive outcomes.