Problem 50 - 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 simplified expression is \( x^2 + 5 \) because it is always positive.

Step 1: Identify the Expression

First, observe the expression under the square root: \( x^4 + 10x^2 + 25 \). Notice that this expression is a quadratic trinomial in terms of \( x^2 \).

Step 2: Factoring the Quadratic Expression

Next, factor the expression inside the square root. The expression \( x^4 + 10x^2 + 25 \) can be rewritten as \( (x^2)^2 + 2 \cdot 5 \cdot x^2 + 5^2 \), which is a perfect square trinomial.

Step 3: Recognizing the Perfect Square

Identify the perfect square trinomial: \( (x^2)^2 + 2 \cdot 5 \cdot x^2 + 5^2 = (x^2 + 5)^2 \).

Step 4: Simplifying the Square Root

Simplify the square root by using the property \( \sqrt{a^2} = |a| \). Thus, \( \sqrt{(x^2 + 5)^2} = |x^2 + 5| \).},{“title”:“Step 5: Absolute Value Consideration”,“description”:“Since all variables are unrestricted, we consider the non-negative nature of squares. Thus, \( |x^2 + 5| = x^2 + 5 \) because \( x^2 + 5 \) is always positive for all real \( x \).

Key Concepts

Quadratic Trinomials

A quadratic trinomial is an expression containing three terms where the highest degree is a square. It often follows the format \(ax^2 + bx + c\). In our example, the expression under the square root is \(x^4 + 10x^2 + 25\), which can be thought of as a quadratic in terms of \(x^2\).

We can rewrite \(x^4\) as \((x^2)^2\), making it easier to identify the quadratic nature. Recognizing the components:

• \((x^2)^2\) is the square term.
• \(10x^2\) is the linear term, represented in terms of \(x^2\).
• \(25\) is the constant.

Understanding how these terms relate to each other is crucial for further simplification strategies.

Perfect Square Trinomials

A perfect square trinomial is a special case of a quadratic trinomial. It can be factored into a binomial squared. Our expression \(x^4 + 10x^2 + 25\) fits this category. It can be expressed as \((x^2 + 5)^2\).

To break it down, we follow these steps:

• Identify the expression pattern that matches \((a+b)^2 = a^2 + 2ab + b^2\).
• Here, \(a = x^2\) and \(b = 5\).
• Verify that \(10x^2\) matches the inner product \(2ab = 10x^2\).

When these conditions are met, the expression is indeed a perfect square trinomial, simplifying our problem significantly.

Absolute Value

Absolute value refers to the distance a number is from zero on the number line, always positive. When simplifying expressions like \(\sqrt{a^2}\), we use absolute value symbols. Thus, \(\sqrt{(x^2 + 5)^2} = |x^2 + 5|\).

The relationship between the square and square root demands consideration of absolute value to maintain mathematical consistency. Since \(x^2 + 5\) is always positive (because it includes a square \(x^2\), and 5), the absolute value function simplifies to

• For any real number \(x\), \(|x^2 + 5| = x^2 + 5\).

There's no need for additional modification as the terms naturally lead to a positive outcome.

Square Roots

A square root extracts the original value squared, simplifying quadratic trinomials seen as perfect squares. For our expression, we already simplified \(x^4 + 10x^2 + 25\) to \((x^2 + 5)^2\).

To find the square root, recognize a key property:

• \(\sqrt{a^2} = |a|\), ensuring non-negative results.

Applying this to \((x^2 + 5)^2\), simplifies to absolute value \(|x^2 + 5|\).

Understanding these rules strengthens foundation skills in algebra, allowing for a smoother transition into more complex algebraic operations.