Given c(x)=5x²+17x-12 and t(x)=x-5/x+2. Find (c+t)(2)

To find \( (c + t)(2) \) for the given functions \( c(x) = 5x^2 + 17x - 12 \) and \( t(x) = \frac{x - 5}{x + 2} \), we need to first compute \( c(2) \) and \( t(2) \), then add the results together.

**Step 1: Calculate \( c(2) \)**

Substitute \( x = 2 \) into \( c(x) \):

\[
c(2) = 5(2)^2 + 17(2) - 12
\]

Calculate the powers and products:

\[
c(2) = 5(4) + 17(2) - 12
\]

\[
c(2) = 20 + 34 - 12
\]

\[
c(2) = 54 - 12
\]

\[
c(2) = 42
\]

**Step 2: Calculate \( t(2) \)**

Substitute \( x = 2 \) into \( t(x) \):

\[
t(2) = \frac{2 - 5}{2 + 2}
\]

Calculate the numerator and denominator:

\[
t(2) = \frac{-3}{4}
\]

**Step 3: Calculate \( (c + t)(2) \)**

Add \( c(2) \) and \( t(2) \):

\[
(c + t)(2) = c(2) + t(2) = 42 + \left( \frac{-3}{4} \right)
\]

Combine the terms:

\[
(c + t)(2) = 42 - \frac{3}{4}
\]

To combine these, convert 42 to a fraction with a common denominator of 4:

\[
42 = \frac{42 \times 4}{4} = \frac{168}{4}
\]

Then add the fractions:

\[
(c + t)(2) = \frac{168}{4} - \frac{3}{4} = \frac{168 - 3}{4} = \frac{165}{4}
\]

So,

\[
(c + t)(2) = \frac{165}{4}
\]

If you need further details or have any questions, please let me know.

Here are 8 related questions for further practice:

1. Find \( (c + t)(3) \).
2. Calculate \( c(0) \).
3. What is \( t(1) \)?
4. Determine the value of \( (c - t)(2) \).
5. Evaluate \( (c \cdot t)(2) \).
6. Find the roots of \( c(x) = 0 \).
7. Simplify \( t(x) \) and find its domain.
8. What is the derivative of \( c(x) \)?

**Tip:** When combining functions, always make sure to compute each function separately at the given point before combining the results.

Learn how to calculate (c + t)(2) where c(x) = 5x² + 17x - 12 and t(x) = (x - 5)/(x + 2). Follow a detailed step-by-step solution involving polynomial and rational function evaluation. Suitable for high school students.

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