https://file.aiquickdraw.com/mathbot/file/1728632624541toyuelah.jpg

The task in the image is to arrange the quadratic expressions in completed square form, corresponding to their respective original quadratic equations. The given quadratics are:

1. \( x^2 - 4x \)
2. \( x^2 - 6x + 6 \)
3. \( 2x^2 - 4x - 3 \)
4. \( 4x^2 - 4x + 2 \)

Their corresponding completed square forms are shown below the quadratics, but we need to match them correctly.

Let's complete the square for each expression:

### 1. \( x^2 - 4x \)
- Step 1: Half of the coefficient of \( x \) is \( -2 \).
- Step 2: Square \( -2 \), which is 4.
- Step 3: Rewrite the expression: \( x^2 - 4x = (x - 2)^2 - 4 \).

So the first expression corresponds to \( (x - 2)^2 - 4 \), which matches with **iv** in the boxed answers.

### 2. \( x^2 - 6x + 6 \)
- Step 1: Half of \( -6 \) is \( -3 \).
- Step 2: Square \( -3 \), which is 9.
- Step 3: Rewrite \( x^2 - 6x + 6 = (x - 3)^2 - 9 + 6 = (x - 3)^2 - 3 \).

So the second expression corresponds to \( (x - 3)^2 - 3 \), which matches with **ii** in the boxed answers.

### 3. \( 2x^2 - 4x - 3 \)
- Step 1: Factor out the 2 from the quadratic terms: \( 2(x^2 - 2x) - 3 \).
- Step 2: Half of \( -2 \) is \( -1 \), and squaring it gives 1.
- Step 3: Add and subtract 1 inside the parentheses: \( 2((x - 1)^2 - 1) - 3 = 2(x - 1)^2 - 2 - 3 = 2(x - 1)^2 - 5 \).

So the third expression corresponds to \( 2(x - 1)^2 - 5 \), which matches with **i** in the boxed answers.

### 4. \( 4x^2 - 4x + 2 \)
- Step 1: Factor out the 4: \( 4(x^2 - x) + 2 \).
- Step 2: Half of \( -1 \) is \( -\frac{1}{2} \), and squaring it gives \( \frac{1}{4} \).
- Step 3: Rewrite: \( 4\left( \left( x - \frac{1}{2} \right)^2 - \frac{1}{4} \right) + 2 = 4\left( x - \frac{1}{2} \right)^2 - 1 + 2 = 4\left( x - \frac{1}{2} \right)^2 + 1 \).

So the fourth expression corresponds to \( (2x - 1)^2 + 1 \), which matches with **iii** in the boxed answers.

### Final matching:
1. **iv** \( (2 - x)^2 - 4 \)
2. **ii** \( (x - 3)^2 - 3 \)
3. **i** \( 2(x - 1)^2 - 5 \)
4. **iii** \( (2x - 1)^2 + 1 \)

Let me know if you need more details or further clarification! Here are five questions you can explore:

1. How do you complete the square for a quadratic expression in general?
2. What are the benefits of converting a quadratic expression into its completed square form?
3. How does factoring affect the process of completing the square when dealing with quadratic expressions like \( 2x^2 \)?
4. Why is the completed square form useful for solving quadratic equations?
5. How does completing the square help in finding the vertex of a parabola?

**Tip**: Completing the square is particularly useful when deriving the quadratic formula or finding the maximum/minimum values of a quadratic function.

Seřaďte výrazy doplněné na úplný čtverec podle pořadí kvadratických trojčlenů: i. x^2 - 4x ii. x^2 - 6x + 6 iii. 2x^2 - 4x - 3 iv. 4x^2 - 4x + 2

Learn how to match quadratic expressions with their corresponding completed square forms, including detailed step-by-step solutions. This problem explores completing the square, factoring, and quadratic expressions, suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation