Math Problem Statement
x² - 4x + 12y + 4 = 0
Solution
We are given the equation:
This is a quadratic equation involving both and . 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 -terms
The -terms in the equation are . Let's complete the square:
Step 2: Substitute into the equation
Now, substitute into the original equation:
Simplify the constants:
Step 3: Solve for
Now, solve for :
Interpretation
This equation represents a parabola that opens downwards. The vertex of this parabola is at , and it is scaled vertically by a factor of .
Would you like further details, or have any specific questions on this? Here are some related questions to consider:
- How can you determine the vertex of a parabola?
- How does completing the square help in identifying the form of a quadratic equation?
- What is the geometric meaning of the parabola's axis of symmetry?
- How does the coefficient affect the shape of the parabola?
- 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