Math Problem Statement

x² - 4x + 12y + 4 = 0

Solution

We are given the equation:

x24x+12y+4=0x^2 - 4x + 12y + 4 = 0

This is a quadratic equation involving both xx and yy. To better understand and analyze this equation, we can attempt to express it in a more standard form.

Step 1: Completing the square for the xx-terms

The xx-terms in the equation are x24xx^2 - 4x. Let's complete the square:

x24x=(x2)24x^2 - 4x = (x - 2)^2 - 4

Step 2: Substitute into the equation

Now, substitute (x2)24(x - 2)^2 - 4 into the original equation:

(x2)24+12y+4=0(x - 2)^2 - 4 + 12y + 4 = 0

Simplify the constants:

(x2)2+12y=0(x - 2)^2 + 12y = 0

Step 3: Solve for yy

Now, solve for yy:

12y=(x2)212y = -(x - 2)^2

y=112(x2)2y = -\frac{1}{12}(x - 2)^2

Interpretation

This equation represents a parabola that opens downwards. The vertex of this parabola is at (2,0)(2, 0), and it is scaled vertically by a factor of 112\frac{1}{12}.

Would you like further details, or have any specific questions on this? Here are some related questions to consider:

  1. How can you determine the vertex of a parabola?
  2. How does completing the square help in identifying the form of a quadratic equation?
  3. What is the geometric meaning of the parabola's axis of symmetry?
  4. How does the coefficient 112-\frac{1}{12} affect the shape of the parabola?
  5. How would you graph this parabola?

Tip: Completing the square is a useful technique not only for solving quadratic equations but also for transforming them into standard forms to identify geometric properties like the vertex and axis of symmetry.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Completing the Square
Parabolas

Formulas

Completing the square formula: x^2 - 4x = (x - 2)^2 - 4
Equation of a parabola: y = -(1/12)(x - 2)^2

Theorems

Quadratic Equation Transformation

Suitable Grade Level

Grades 9-12