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 becomes , let’s go step by step and see how the simplification happens.
Step-by-Step Simplification:
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Starting expression:
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Recognizing a quadratic form: The expression inside the square root is . This is a quadratic in terms of . Let’s rewrite it:
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Check if it is a perfect square trinomial: Recall the expansion of : Here:
Substitute these values: Simplify:
Thus, the original expression is indeed equal to .
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Simplifying the square root: Now substitute this back into the square root:
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Simplify the square root of a square: The square root of a square is , the absolute value of the expression:
Since for all real values of , the absolute value is unnecessary, so:
Where did go?
The term is part of the perfect square trinomial . It is not "gone"; rather, it is embedded in the expanded form of .
Let me know if you'd like further clarifications or have any specific questions!
Here are 5 related questions:
- What is a perfect square trinomial, and how do we recognize one?
- How does the square root simplify for perfect square expressions?
- Why does simplify to ?
- What are other examples of expressions that can be written as perfect square trinomials?
- Can be factored differently than ?
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