How to Simplify Quadratic Functions: Simplifying $ g(u) $ and Finding $ g(x^2 - 1) $

Understanding how to simplify quadratic functions is essential for solving equations and working with algebraic expressions efficiently. In this article, we’ll walk through simplifying a quadratic function $ g(u) $, then use it to compute $ g(x^2 - 1) $, demonstrating step-by-step simplification. This method can help simplify complex expressions in algebra and calculus.

Understanding the Context

Step 1: Simplify the Quadratic Function $ g(u) $

Let’s begin by simplifying the function:

$$
g(u) = (u - 3)^2 + 6(u - 3) + 5
$$

We expand each term carefully:

ight) + 5 $. Simplify: $ g(u) = (u - 3)^2 + 6(u - 3) + 5 = u^2 - 6u + 9 + 6u - 18 + 5 = u^2 - 4 $. Thus, $ g(x^2 - 1) = (x^2 - 1)^2 - 4 = x^4 - 2x^2 + 1 - 4 = x^4 - 2x^2 - 3 $. Final answer: $oxed{x^4 - 2x^2 - 3}$

Key Insights

  • Expand $ (u - 3)^2 $:
    $$
    (u - 3)^2 = u^2 - 6u + 9
    $$

  • Expand $ 6(u - 3) $:
    $$
    6(u - 3) = 6u - 18
    $$

  • Add the constant 5.

Now combine all terms:

$$
g(u) = (u^2 - 6u + 9) + (6u - 18) + 5
$$

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Final Thoughts

Group like terms:

$$
u^2 + (-6u + 6u) + (9 - 18 + 5) = u^2 - 4
$$

So, the simplified function is:

$$
g(u) = u^2 - 4
$$

Step 2: Substitute $ u = x^2 - 1 $ into $ g(u) $

Now, use $ g(u) = u^2 - 4 $ to find $ g(x^2 - 1) $:

$$
g(x^2 - 1) = (x^2 - 1)^2 - 4
$$

Expand $ (x^2 - 1)^2 $:

$$
(x^2 - 1)^2 = x^4 - 2x^2 + 1
$$