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: $oxed{x^4 - 2x^2 - 3}$
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.
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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Expand $ (u - 3)^2 $: $$ (u - 3)^2 = u^2 - 6u + 9 $$
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Expand $ 6(u - 3) $: $$ 6(u - 3) = 6u - 18 $$
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Add the constant 5.
Now combine all terms:
$$ g(u) = (u^2 - 6u + 9) + (6u - 18) + 5 $$
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 $:
$$ (x^2 - 1)^2 = x^4 - 2x^2 + 1 $$
Subtract 4:
$$ g(x^2 - 1) = x^4 - 2x^2 + 1 - 4 = x^4 - 2x^2 - 3 $$
Final Result
After simplifying step by step, we find:
$$ oxed{g(x^2 - 1) = x^4 - 2x^2 - 3} $$
Why This Matters
Simplifying functions like $ g(u) $ helps streamline expressions before substitution. In this case, simplifying $ g(u) $ first made substitution easier and reduced errors. This technique applies to many algebraic problems, making equations cleaner and easier to solve.
If you often work with quadratic functions and substitutions, mastering algebraic simplification will save time and boost accuracy. Try simplifying your own functions today—you’ll see how much clearer the math becomes!
