Solve for C C(t)=t^2-2t-4

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.

1. Division of Algebraic Expressions: When dividing algebraic expressions, terms with the same base can be divided, and their exponents subtracted. For example, $\frac{t^2}{t}=t^{2-1}=t$.

2. Simplification: This process involves canceling out common factors in the numerator and denominator, which is a fundamental aspect of algebraic manipulation.

3. 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.

4. Algebraic Identity: $t^1=t$. Any non-zero number raised to the power of 1 is the number itself.

5. 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.