Solve for C C(t)=t^2-2t-4
The given problem is a mathematics question asking to find the solution for the variable 'C' given the function C(t) = t^2 - 2t - 4. It requires you to manipulate the function or perform operations with respect to the variable 't' in order to express 'C' as a function of time 't'. This might involve simplifying the equation or finding the roots of the quadratic equation, depending on the context of the question.
$C \left(\right. t \left.\right) = t^{2} - 2 t - 4$
Answer
Solution:
Step 1 Isolate $C$ by dividing each term of $C(t) = t^2 - 2t - 4$ by $t$ to get $\frac{C(t)}{t} = \frac{t^2}{t} - \frac{2t}{t} - \frac{4}{t}$.
Step 2 Simplify the equation by working on the left side first.
Step 2.1 Eliminate the common $t$ in the numerator and denominator.
Step 2.1.1 Remove the $t$ from $C(t)$ to get $C = \frac{t^2}{t} - \frac{2t}{t} - \frac{4}{t}$.
Step 2.1.2 Recognize that $C$ divided by $1$ remains $C$, so $C = \frac{t^2}{t} - \frac{2t}{t} - \frac{4}{t}$.
Step 3 Now, simplify the right side of the equation.
Step 3.1 Handle each term individually.
Step 3.1.1 Focus on the $t^2$ term.
Step 3.1.1.1 Extract $t$ from $t^2$ to get $C = \frac{t \cdot t}{t} - \frac{2t}{t} - \frac{4}{t}$.
Step 3.1.1.2 Proceed to cancel out the common factors.
Step 3.1.1.2.1 Recognize that $t$ raised to the power of $1$ is simply $t$, so $C = \frac{t \cdot t}{t^1} - \frac{2t}{t} - \frac{4}{t}$.
Step 3.1.1.2.2 Factor $t$ out of $t^1$ to get $C = \frac{t \cdot t}{t \cdot 1} - \frac{2t}{t} - \frac{4}{t}$.
Step 3.1.1.2.3 Cancel the $t$ in the numerator and denominator to simplify.
Step 3.1.1.2.4 Rewrite the simplified term as $C = \frac{t}{1} - \frac{2t}{t} - \frac{4}{t}$.
Step 3.1.1.2.5 Divide $t$ by $1$ to get $C = t - \frac{2t}{t} - \frac{4}{t}$.
Step 3.1.2 Address the $-2t$ term.
Step 3.1.2.1 Remove the common $t$ to get $C = t - \frac{2 \cancel{t}}{\cancel{t}} - \frac{4}{t}$.
Step 3.1.2.2 Recognize that $-2$ divided by $1$ is $-2$, so $C = t - 2 - \frac{4}{t}$.
Step 3.1.3 Position the negative sign in front of the fraction for the last term.
$C = t - 2 - \frac{4}{t}$.
Knowledge Notes:
The problem-solving process involves simplifying an algebraic expression by dividing each term by a common variable, in this case, $t$. The steps taken are systematic, ensuring that each term is simplified individually.
Division of Algebraic Expressions: When dividing algebraic expressions, terms with the same base can be divided, and their exponents subtracted. For example, $\frac{t^2}{t} = t^{2-1} = t$.
Simplification: This process involves canceling out common factors in the numerator and denominator, which is a fundamental aspect of algebraic manipulation.
Negative Signs and Fractions: When dealing with negative signs in fractions, it's important to correctly position the sign to maintain the expression's integrity.
Algebraic Identity: $t^1 = t$. Any non-zero number raised to the power of 1 is the number itself.
Division by One: Dividing any number by one leaves the number unchanged, which is a basic arithmetic rule.
The solution process uses these concepts to simplify the given function $C(t) = t^2 - 2t - 4$ to its simplest form by dividing each term by $t$ and simplifying the resulting expression.
