Math Problem Statement

Transform each quadratic function in the form y=ax²+ bc+c 1)y-x+5=(x-2)²

Solution

Let's transform the given quadratic equation yx+5=(x2)2y - x + 5 = (x - 2)^2 into the standard form y=ax2+bx+cy = ax^2 + bx + c.

Step 1: Expand the squared term

First, expand the right-hand side of the equation: (x2)2=(x2)(x2)=x24x+4(x - 2)^2 = (x - 2)(x - 2) = x^2 - 4x + 4 Now, the equation becomes: yx+5=x24x+4y - x + 5 = x^2 - 4x + 4

Step 2: Simplify the equation

Next, move all terms that are not related to yy to the right-hand side: y=x24x+4+x5y = x^2 - 4x + 4 + x - 5

Step 3: Combine like terms

Simplify the expression on the right-hand side: y=x24x+x+45y = x^2 - 4x + x + 4 - 5 y=x23x1y = x^2 - 3x - 1

Final Answer:

The equation in the form y=ax2+bx+cy = ax^2 + bx + c is: y=x23x1y = x^2 - 3x - 1

Do you need any details or have questions?

Here are 5 questions that could extend the understanding:

  1. How do you expand binomials like (x2)2(x-2)^2?
  2. What is the significance of combining like terms in this transformation?
  3. Can you graph this quadratic function and interpret its key features (vertex, intercepts)?
  4. How would completing the square apply to this equation?
  5. What does the coefficient a=1a = 1 tell you about the shape of the parabola?

Tip: Always check for like terms when simplifying quadratic equations to avoid errors.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Expanding Binomials
Simplification

Formulas

Standard form of a quadratic equation: y = ax^2 + bx + c

Theorems

None directly involved; focuses on algebraic manipulation and simplification

Suitable Grade Level

Grades 8-10