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: $oxedx^4 - 2x^2 - 3$ - Coaching Toolbox
How to Simplify Quadratic Functions: Simplifying $ g(u) $ and Finding $ g(x^2 - 1) $
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:
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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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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 $:
$$
(x^2 - 1)^2 = x^4 - 2x^2 + 1
$$
