Huff Turnbull
06/05/2024 · Senior High School
\( \left. \begin{array} { l } { f ( x ) = 3 x ^ { 3 } - 10 x ^ { 2 } + 7 x + 50 } \\ { g ( x ) = 12 x ^ { 2 } - 14 x + 70 } \\ { f ( x ) + g ( x ) } \\ { f ( x ) - g ( x ) } \\ { f ( x ) = g ( x ) } \end{array} \right. \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
Let's analyze the functions \( f(x) \) and \( g(x) \) given in the problem.
1. **Define the functions:**
\[
f(x) = 3x^3 - 10x^2 + 7x + 50
\]
\[
g(x) = 12x^2 - 14x + 70
\]
2. **Calculate \( f(x) + g(x) \):**
\[
f(x) + g(x) = (3x^3 - 10x^2 + 7x + 50) + (12x^2 - 14x + 70)
\]
Combine like terms:
\[
= 3x^3 + (-10x^2 + 12x^2) + (7x - 14x) + (50 + 70)
\]
\[
= 3x^3 + 2x^2 - 7x + 120
\]
3. **Calculate \( f(x) - g(x) \):**
\[
f(x) - g(x) = (3x^3 - 10x^2 + 7x + 50) - (12x^2 - 14x + 70)
\]
Combine like terms:
\[
= 3x^3 + (-10x^2 - 12x^2) + (7x + 14x) + (50 - 70)
\]
\[
= 3x^3 - 22x^2 + 21x - 20
\]
4. **Set \( f(x) = g(x) \):**
To find the values of \( x \) where \( f(x) = g(x) \), we set the two functions equal to each other:
\[
3x^3 - 10x^2 + 7x + 50 = 12x^2 - 14x + 70
\]
Rearranging gives:
\[
3x^3 - 10x^2 - 12x^2 + 7x + 14x + 50 - 70 = 0
\]
\[
3x^3 - 22x^2 + 21x - 20 = 0
\]
Now we have the following results:
- \( f(x) + g(x) = 3x^3 + 2x^2 - 7x + 120 \)
- \( f(x) - g(x) = 3x^3 - 22x^2 + 21x - 20 \)
- The equation \( f(x) = g(x) \) simplifies to \( 3x^3 - 22x^2 + 21x - 20 = 0 \).
If you need further analysis or specific solutions for the equation \( f(x) = g(x) \), please let me know!
Quick Answer
- \( f(x) + g(x) = 3x^3 + 2x^2 - 7x + 120 \)
- \( f(x) - g(x) = 3x^3 - 22x^2 + 21x - 20 \)
- \( f(x) = g(x) \) simplifies to \( 3x^3 - 22x^2 + 21x - 20 = 0 \)
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