We are given the equation:

\[
x^2 - 4x + 12y + 4 = 0
\]

This is a quadratic equation involving both \(x\) and \(y\). 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 \(x\)-terms
The \(x\)-terms in the equation are \(x^2 - 4x\). Let's complete the square:

\[
x^2 - 4x = (x - 2)^2 - 4
\]

### Step 2: Substitute into the equation
Now, substitute \((x - 2)^2 - 4\) into the original equation:

\[
(x - 2)^2 - 4 + 12y + 4 = 0
\]

Simplify the constants:

\[
(x - 2)^2 + 12y = 0
\]

### Step 3: Solve for \(y\)
Now, solve for \(y\):

\[
12y = -(x - 2)^2
\]

\[
y = -\frac{1}{12}(x - 2)^2
\]

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

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 \(-\frac{1}{12}\) 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.

This page explains how to solve the quadratic equation x² - 4x + 12y + 4 by completing the square. Learn how to transform the equation into a standard form and interpret the resulting parabola. Suitable for students in Grades 9-12, with a focus on parabolas and quadratic transformations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation