Question
Given the function \( p(c)=c^{2}+c \), complete the following. (a) Evaluate \( p(-5) \). (b) Solve \( p(c)=2 \). (Enter your answers as a comma-separated list.) \( c=\square \)
Ask by Dawson Fowler. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
**(a)** \( p(-5) = 20 \).
**(b)** \( c = -2, 1 \).
Solution
**(a)** We substitute \( c = -5 \) into the function:
\[
p(-5) = (-5)^2 + (-5) = 25 - 5 = 20.
\]
**(b)** To solve \( p(c) = 2 \) we set up the equation:
\[
c^2 + c = 2.
\]
Subtract 2 from both sides to obtain:
\[
c^2 + c - 2 = 0.
\]
We factor the quadratic by finding two numbers that multiply to \(-2\) and add to \(1\). These numbers are \(2\) and \(-1\):
\[
c^2 + c - 2 = (c + 2)(c - 1) = 0.
\]
Setting each factor equal to zero gives:
\[
c + 2 = 0 \quad \Rightarrow \quad c = -2,
\]
\[
c - 1 = 0 \quad \Rightarrow \quad c = 1.
\]
Thus, the solutions are:
\[
c = -2, \, 1.
\]
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Simplify this solution Beyond the Answer
To evaluate \( p(-5) \), substitute \(-5\) into the function: \[ p(-5) = (-5)^{2} + (-5) = 25 - 5 = 20. \] Now, for solving \( p(c)=2 \), set up the equation with the function: \[ c^{2} + c = 2. \] Rearranging it gives: \[ c^{2} + c - 2 = 0. \] Now, factor the quadratic: \[ (c - 1)(c + 2) = 0. \] Setting each factor to zero gives: \[ c - 1 = 0 \quad \Rightarrow \quad c = 1, \] \[ c + 2 = 0 \quad \Rightarrow \quad c = -2. \] Thus, the solutions are: \( c = 1, -2 \).
