Math Problem Statement

how do √x^4+4x^2+4 become √(x^2+2)^2 and where do 4x^2 go

Solution

To understand how x4+4x2+4\sqrt{x^4 + 4x^2 + 4} becomes (x2+2)2\sqrt{(x^2 + 2)^2}, let’s go step by step and see how the simplification happens.


Step-by-Step Simplification:

  1. Starting expression: x4+4x2+4\sqrt{x^4 + 4x^2 + 4}

  2. Recognizing a quadratic form: The expression inside the square root is x4+4x2+4x^4 + 4x^2 + 4. This is a quadratic in terms of x2x^2. Let’s rewrite it: x4+4x2+4=(x2)2+4x2+22x^4 + 4x^2 + 4 = (x^2)^2 + 4x^2 + 2^2

  3. Check if it is a perfect square trinomial: Recall the expansion of (a+b)2(a + b)^2: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2 Here:

    • a=x2a = x^2
    • b=2b = 2

    Substitute these values: (x2+2)2=(x2)2+2(x2)(2)+22(x^2 + 2)^2 = (x^2)^2 + 2(x^2)(2) + 2^2 Simplify: (x2+2)2=x4+4x2+4(x^2 + 2)^2 = x^4 + 4x^2 + 4

    Thus, the original expression x4+4x2+4x^4 + 4x^2 + 4 is indeed equal to (x2+2)2(x^2 + 2)^2.

  4. Simplifying the square root: Now substitute this back into the square root: x4+4x2+4=(x2+2)2\sqrt{x^4 + 4x^2 + 4} = \sqrt{(x^2 + 2)^2}

  5. Simplify the square root of a square: The square root of a square (x2+2)2\sqrt{(x^2 + 2)^2} is x2+2|x^2 + 2|, the absolute value of the expression: x4+4x2+4=x2+2\sqrt{x^4 + 4x^2 + 4} = |x^2 + 2|

    Since x2+20x^2 + 2 \geq 0 for all real values of xx, the absolute value is unnecessary, so: x4+4x2+4=x2+2\sqrt{x^4 + 4x^2 + 4} = x^2 + 2


Where did 4x24x^2 go?

The term 4x24x^2 is part of the perfect square trinomial x4+4x2+4=(x2+2)2x^4 + 4x^2 + 4 = (x^2 + 2)^2. It is not "gone"; rather, it is embedded in the expanded form of (x2+2)2(x^2 + 2)^2.


Let me know if you'd like further clarifications or have any specific questions!

Here are 5 related questions:

  1. What is a perfect square trinomial, and how do we recognize one?
  2. How does the square root simplify for perfect square expressions?
  3. Why does (x2+2)2\sqrt{(x^2 + 2)^2} simplify to x2+2x^2 + 2?
  4. What are other examples of expressions that can be written as perfect square trinomials?
  5. Can x4+4x2+4x^4 + 4x^2 + 4 be factored differently than (x2+2)2(x^2 + 2)^2?

Tip: Always check for patterns like perfect squares when simplifying square roots or factoring expressions!

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Square Roots
Perfect Squares

Formulas

Perfect Square Trinomial: (a + b)^2 = a^2 + 2ab + b^2
Square Root of a Square: √(a^2) = |a|

Theorems

Perfect Square Trinomial
Properties of Square Roots

Suitable Grade Level

Grades 9-11