To complete the square for the expression \( 5 - 4t - t^2 \), follow these steps:

### Step 1: Reorder the terms
First, reorder the terms so that the quadratic term (\(t^2\)) comes first, followed by the linear term (\(t\)) and then the constant:
\[
-t^2 - 4t + 5
\]

### Step 2: Factor out the coefficient of \(t^2\)
Factor out the negative sign from the first two terms:
\[
- (t^2 + 4t) + 5
\]

### Step 3: Complete the square inside the parentheses
To complete the square, take the coefficient of \(t\), which is 4, divide it by 2, and then square the result:
\[
\left(\frac{4}{2}\right)^2 = 2^2 = 4
\]
Add and subtract this square inside the parentheses:
\[
- \left(t^2 + 4t + 4 - 4\right) + 5
\]
This simplifies to:
\[
- \left((t + 2)^2 - 4\right) + 5
\]

### Step 4: Distribute the negative sign
Distribute the negative sign across the terms inside the parentheses:
\[
- (t + 2)^2 + 4 + 5
\]

### Step 5: Combine like terms
Finally, combine the constants:
\[
- (t + 2)^2 + 9
\]

### Final Answer:
The expression \( 5 - 4t - t^2 \) can be written in its completed square form as:
\[
-(t + 2)^2 + 9
\]

Would you like any additional details? Here are five related questions:

1. How does completing the square help in solving quadratic equations?
2. Can you explain how the vertex form of a quadratic equation relates to completing the square?
3. What is the significance of the coefficient of \(t^2\) when completing the square?
4. How can you use the completed square form to find the maximum or minimum value of a quadratic expression?
5. What are some common mistakes to avoid when completing the square?

**Tip:** Completing the square is particularly useful when working with quadratic functions, as it allows you to easily identify the vertex and rewrite the function in a more usable form.

Learn how to complete the square for the quadratic expression 5 - 4t - t^2 with a detailed step-by-step solution. Understand the significance of the vertex form of a quadratic equation and its application in solving such expressions. Suitable for high school students studying quadratic equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation