(a^(2)+1)^(2)+(a^(2)+5)^(2)-4(a^(2)+3)^(2)

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression: (a^(2)+1)^(2)+(a^(2)+5)^(2)-4(a^(2)+3)^(2)

This article explores the simplification of the algebraic expression: (a^(2)+1)^(2)+(a^(2)+5)^(2)-4(a^(2)+3)^(2). We will break down the process step-by-step to arrive at a simplified form.

Expanding the Squares

First, we expand the squares using the formula (x + y)^2 = x^2 + 2xy + y^2:

• (a^(2)+1)^(2) = a^4 + 2a^2 + 1
• (a^(2)+5)^(2) = a^4 + 10a^2 + 25
• (a^(2)+3)^(2) = a^4 + 6a^2 + 9

Now, let's substitute these expanded forms back into the original expression:

(a^4 + 2a^2 + 1) + (a^4 + 10a^2 + 25) - 4(a^4 + 6a^2 + 9)

Combining Like Terms

Next, we distribute the -4 and combine like terms:

a^4 + 2a^2 + 1 + a^4 + 10a^2 + 25 - 4a^4 - 24a^2 - 36

This simplifies to:

-2a^4 - 12a^2 - 10

Final Simplified Form

Therefore, the simplified form of the expression (a^(2)+1)^(2)+(a^(2)+5)^(2)-4(a^(2)+3)^(2) is -2a^4 - 12a^2 - 10.

This simplification demonstrates the power of algebraic manipulation in reducing complex expressions to their simplest forms.