Transform this quadratic function in the form y=ax²+bc+c

1)y-x+5=(x-2)²
2)x+5=(x+1)²-y
3)4x=x²+4-y
4)y-x=x²-x+1
5)4x-y=x+5

Let's transform each quadratic equation into the standard form \( y = ax^2 + bx + c \).

### 1) \( y - x + 5 = (x - 2)^2 \)

Expand \( (x - 2)^2 \):
\[
(x - 2)^2 = x^2 - 4x + 4
\]
So the equation becomes:
\[
y - x + 5 = x^2 - 4x + 4
\]
Solve for \( y \):
\[
y = x^2 - 4x + 4 + x - 5
\]
Simplify:
\[
y = x^2 - 3x - 1
\]
Thus, in the form \( y = ax^2 + bx + c \), we have:
\[
y = x^2 - 3x - 1
\]

### 2) \( x + 5 = (x + 1)^2 - y \)

Expand \( (x + 1)^2 \):
\[
(x + 1)^2 = x^2 + 2x + 1
\]
So the equation becomes:
\[
x + 5 = x^2 + 2x + 1 - y
\]
Solve for \( y \):
\[
y = x^2 + 2x + 1 - x - 5
\]
Simplify:
\[
y = x^2 + x - 4
\]
Thus, in the form \( y = ax^2 + bx + c \), we have:
\[
y = x^2 + x - 4
\]

### 3) \( 4x = x^2 + 4 - y \)

Solve for \( y \):
\[
y = x^2 + 4 - 4x
\]
Rearrange:
\[
y = x^2 - 4x + 4
\]
Thus, in the form \( y = ax^2 + bx + c \), we have:
\[
y = x^2 - 4x + 4
\]

### 4) \( y - x = x^2 - x + 1 \)

Solve for \( y \):
\[
y = x^2 - x + 1 + x
\]
Simplify:
\[
y = x^2 + 1
\]
Thus, in the form \( y = ax^2 + bx + c \), we have:
\[
y = x^2 + 1
\]

### 5) \( 4x - y = x + 5 \)

Solve for \( y \):
\[
y = 4x - x - 5
\]
Simplify:
\[
y = 3x - 5
\]
Thus, in the form \( y = ax^2 + bx + c \), we have:
\[
y = 0x^2 + 3x - 5
\]

### Final Answers:
1. \( y = x^2 - 3x - 1 \)
2. \( y = x^2 + x - 4 \)
3. \( y = x^2 - 4x + 4 \)
4. \( y = x^2 + 1 \)
5. \( y = 3x - 5 \)

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Would you like further details or explanations on any of the steps?

Here are 5 related questions to consider:
1. How do you complete the square for any quadratic equation?
2. What are the general forms of a quadratic equation?
3. How can you determine the vertex from a quadratic equation?
4. How does factoring help in solving quadratic equations?
5. What are the properties of a parabola described by \( y = ax^2 + bx + c \)?

**Tip:** Always simplify and rearrange terms carefully when transforming equations to avoid mistakes.

Transform this quadratic function in the form y = ax² + bx + c:
1) y - x + 5 = (x - 2)²
2) x + 5 = (x + 1)² - y
3) 4x = x² + 4 - y
4) y - x = x² - x + 1
5) 4x - y = x + 5

Learn how to transform quadratic equations into the standard form y = ax² + bx + c. This guide provides step-by-step solutions for various examples, explaining the use of algebraic concepts such as expanding binomials and completing the square. Suitable for students in Grades 8-10.

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