Complete the Square for 5 - 4t + t^2: Step-by-Step

Let's complete the square for the quadratic expression \( 5 - 4t + t^2 \). We'll go through this step by step.

### Expression: 
\[
t^2 - 4t + 5
\]
(Note: I rearranged the terms in descending powers of \( t \), though this doesn't change the math.)

### Step 1: Move the constant to the other side
To focus on completing the square, first deal with the \( t^2 \) and \( t \)-terms. So, separate the constant term:
\[
t^2 - 4t = -5
\]

### Step 2: Complete the square
To complete the square on \( t^2 - 4t \), follow this procedure:
1. Take the coefficient of \( t \) (which is \( -4 \)), divide by 2, and then square it:
   \[
   \left( \frac{-4}{2} \right)^2 = (-2)^2 = 4
   \]

2. Add this value to both sides of the equation:
   \[
   t^2 - 4t + 4 = -5 + 4
   \]

### Step 3: Simplify both sides
Now simplify both sides of the equation:
\[
(t - 2)^2 = -1
\]

### Step 4: Solve or rewrite (optional)
The quadratic expression can now be written as:
\[
t^2 - 4t + 5 = (t - 2)^2 + 1
\]

### Final Answer:
The completed square form of \( 5 - 4t + t^2 \) is:
\[
(t - 2)^2 + 1
\]

This is the desired form!

Would you like more details on completing the square, or do you have any questions?

---

Here are 5 related questions you might find helpful:
1. How can completing the square help solve quadratic equations?
2. What is the quadratic formula, and how is it derived from completing the square?
3. How do you complete the square when the coefficient of \( t^2 \) is not 1?
4. Can you complete the square for \( 3t^2 - 6t + 2 \)?
5. What is the geometric interpretation of completing the square?

**Tip**: Completing the square is a helpful technique to convert a quadratic expression into vertex form, which makes it easier to analyze the graph of the function.

Learn how to complete the square for the quadratic expression 5 - 4t + t^2 with a detailed step-by-step guide. This method helps rewrite quadratic expressions into perfect square form, suitable for Grades 9-11 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation