Math Problem Statement

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

Solution

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. x24xx^2 - 4x
  2. x26x+6x^2 - 6x + 6
  3. 2x24x32x^2 - 4x - 3
  4. 4x24x+24x^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. x24xx^2 - 4x

  • Step 1: Half of the coefficient of xx is 2-2.
  • Step 2: Square 2-2, which is 4.
  • Step 3: Rewrite the expression: x24x=(x2)24x^2 - 4x = (x - 2)^2 - 4.

So the first expression corresponds to (x2)24(x - 2)^2 - 4, which matches with iv in the boxed answers.

2. x26x+6x^2 - 6x + 6

  • Step 1: Half of 6-6 is 3-3.
  • Step 2: Square 3-3, which is 9.
  • Step 3: Rewrite x26x+6=(x3)29+6=(x3)23x^2 - 6x + 6 = (x - 3)^2 - 9 + 6 = (x - 3)^2 - 3.

So the second expression corresponds to (x3)23(x - 3)^2 - 3, which matches with ii in the boxed answers.

3. 2x24x32x^2 - 4x - 3

  • Step 1: Factor out the 2 from the quadratic terms: 2(x22x)32(x^2 - 2x) - 3.
  • Step 2: Half of 2-2 is 1-1, and squaring it gives 1.
  • Step 3: Add and subtract 1 inside the parentheses: 2((x1)21)3=2(x1)223=2(x1)252((x - 1)^2 - 1) - 3 = 2(x - 1)^2 - 2 - 3 = 2(x - 1)^2 - 5.

So the third expression corresponds to 2(x1)252(x - 1)^2 - 5, which matches with i in the boxed answers.

4. 4x24x+24x^2 - 4x + 2

  • Step 1: Factor out the 4: 4(x2x)+24(x^2 - x) + 2.
  • Step 2: Half of 1-1 is 12-\frac{1}{2}, and squaring it gives 14\frac{1}{4}.
  • Step 3: Rewrite: 4((x12)214)+2=4(x12)21+2=4(x12)2+14\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 (2x1)2+1(2x - 1)^2 + 1, which matches with iii in the boxed answers.

Final matching:

  1. iv (2x)24(2 - x)^2 - 4
  2. ii (x3)23(x - 3)^2 - 3
  3. i 2(x1)252(x - 1)^2 - 5
  4. iii (2x1)2+1(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 2x22x^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.

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Math Problem Analysis

Mathematical Concepts

Completing the Square
Quadratic Expressions
Factoring

Formulas

Completing the square: ax^2 + bx + c = a(x-h)^2 + k
Quadratic expression

Theorems

Method of Completing the Square
Factoring Quadratics

Suitable Grade Level

Grades 9-12