Problem 50 - Elementary and Intermediate Algebra

Question

Simplify each expression. Assume that all variables are unrestricted and use absolute value symbols when necessary. See Example 2. $$ \sqrt{x^{4}+10 x^{2}+25} $$

Step-by-Step Solution

Answer

The expression simplifies to \( x^2 + 5 \).

Step 1: Identify as a Perfect Square Trinomial

The given expression inside the square root is \( x^4 + 10x^2 + 25 \). We recognize this as a quadratic expression in terms of \( x^2 \).

Step 2: Rewrite the Trinomial

Notice that \( x^4 = (x^2)^2 \), \( 10x^2 = 2 \cdot 5 \cdot x^2 \), and \( 25 = 5^2 \). Recognize that the expression is of the form \( a^2 + 2ab + b^2 \), which factors into \( (a + b)^2 \).

Step 3: Factor the Expression

We can factor \( x^4 + 10x^2 + 25 \) as \( (x^2 + 5)^2 \) because it fits the pattern \( a^2 + 2ab + b^2 \) with \( a = x^2 \) and \( b = 5 \).

Step 4: Simplify the Square Root

Now that we can express the trinomial as \( (x^2 + 5)^2 \), taking the square root yields \( \sqrt{(x^2 + 5)^2} = |x^2 + 5| \). Since \( x^2 + 5 \) is always non-negative (as squares are always positive and we've added a positive number), the absolute value function can be removed.

Step 5: Final Expression

Thus, the simplified form of the expression is \( x^2 + 5 \).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special form of a quadratic expression that can be rewritten as the square of a binomial. The general form is \( a^2 + 2ab + b^2 \), which can be factored into \((a + b)^2\). This pattern allows us to easily identify and simplify the expression.
In the given problem, we started with the expression \( x^4 + 10x^2 + 25 \). By comparing it to the perfect square trinomial formula, we identified:

• \( a^2 = x^4 \), making \( a = x^2 \)
• \( 2ab = 10x^2 \)
• \( b^2 = 25 \), so \( b = 5 \)

Hence, the expression becomes \((x^2 + 5)^2\). Understanding this pattern helps in simplifying and recognizing similar expressions quickly.

Absolute Value

The absolute value represents the distance of a number or expression from zero on a number line, without considering direction. It's always non-negative.
When simplifying square root expressions of the form \( \sqrt{(expression)^2} \), it yields the absolute value of the expression. So for an expression like \( \sqrt{(x^2 + 5)^2} \), this becomes \( |x^2 + 5| \), showing the magnitude or distance from zero of \( x^2 + 5 \).
However, since \( x^2 \) (a square) and 5 (a positive number) together always produce a non-negative result, the expression inside the absolute value is always non-negative. Therefore, we can safely discard the absolute value symbols in this case, simplifying the expression directly to \( x^2 + 5 \). This step is essential when simplifying expressions involving squares and square roots.

Quadratic Expressions

Quadratic expressions are polynomial expressions where the highest degree of the variable is 2. They generally take the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.
In the context of our exercise, the expression \( x^4 + 10x^2 + 25 \) is a quadratic expression in terms of \( x^2 \), because \( x^4 \) is \((x^2)^2\). Recognizing this allowed us to rewrite and simplify it by identifying it as a perfect square trinomial.

• Let \( y = x^2 \), then the expression becomes \( y^2 + 10y + 25 \).
• This is now an identifiable quadratic expression which simplifies to \((y + 5)^2\).
• Substituting \( y = x^2 \) back gives \((x^2 + 5)^2\).

This approach of using quadratic forms is a powerful technique for simplifying higher-degree expressions.