4(x^2 - 8x) - 9(y^2 - 6y) = -115. - United Radiology

["# Understanding the Equation (4(x^2 - 8x) - 9(y^2 - 6y) = -115): A Comprehensive Guide", "The equation (4(x^2 - 8x) - 9(y^2 - 6y) = -115) might appear complex at first glance, but it holds key insights into conic sections—specifically, a hyperbola—through its structure and transformation into standard form. In this SEO-rich article, we break down the equation step-by-step, explore its geometric implications, and explain how mastering it can boost your understanding of conic equations.", "## Equation Breakdown: Rewriting in Standard Form", "To interpret this equation, we begin by expanding and rearranging it into a more recognizable mathematical form.", "### Step 1: Expand the quadratic terms
\nStart by distributing the coefficients:", "[
\n4(x^2 - 8x) - 9(y^2 - 6y) = -115
\n]
\n[
\n4x^2 - 32x - 9y^2 + 54y = -115
\n]", "### Step 2: Move constant to the right side
\nAdd 115 to both sides:", "[
\n4x^2 - 32x - 9y^2 + 54y + 115 = 0
\n]", "### Step 3: Group x and y terms
\nGroup the (x)-terms and (y)-terms together:", "[
\n(4x^2 - 32x) + (-9y^2 + 54y) + 115 = 0
\n]", "### Step 4: Factor coefficients of quadratic terms
\nFactor out common coefficients:", "[
\n4(x^2 - 8x) - 9(y^2 - 6y) + 115 = 0
\n]", "Note: To convert to a standard hyperbola form, we complete the square for both groups.", "---", "### Completing the Square", "We complete the square for (x^2 - 8x) and (y^2 - 6y) separately.", "For (x):
\n(x^2 - 8x) → take half of (-8), which is (-4), square it → (16)", "[
\nx^2 - 8x = (x - 4)^2 - 16
\n]", "Multiply by 4:
\n[
\n4(x^2 - 8x) = 4[(x - 4)^2 - 16] = 4(x - 4)^2 - 64
\n]", "For (y):
\n(y^2 - 6y) → half of (-6) is (-3), square → (9)", "[
\ny^2 - 6y = (y - 3)^2 - 9
\n]", "Multiply by (-9):
\n[
\n-9(y^2 - 6y) = -9[(y - 3)^2 - 9] = -9(y - 3)^2 + 81
\n]", "---", "### Substitute back and simplify", "Replace in the original equation:", "[
\n[4(x - 4)^2 - 64] + [-9(y - 3)^2 + 81] + 115 = 0
\n]", "Combine constants:", "[
\n4(x - 4)^2 - 9(y - 3)^2 - 64 + 81 + 115 = 0
\n]
\n[
\n4(x - 4)^2 - 9(y - 3)^2 + 132 = 0
\n]", "Move constant to the right:", "[
\n4(x - 4)^2 - 9(y - 3)^2 = -132
\n]", "Divide entire equation by (-132) to normalize to standard hyperbola form:", "[
\n\frac{4(x - 4)^2}{-132} - \frac{9(y - 3)^2}{-132} = 1
\n]
\n[
\n-\frac{(x - 4)^2}{33} + \frac{(y - 3)^2}{\frac{132}{9}} = 1
\n]", "Simplify denominators:", "[
\n\frac{(y - 3)^2}{\frac{44}{3}} - \frac{(x - 4)^2}{33} = 1
\n]", "This is the standard form of a hyperbola opening vertically centered at ((4, 3)).", "---", "## Geometric Interpretation: What Does This Hyperbola Represent?", "This equation describes a hyperbola, defined as the locus of points where the difference of distances from two fixed points (the foci) is constant. In standard form:", "[
\n\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1
\n]", "- Center: ((h, k) = (4, 3))
\n- Vertical transverse axis (opens up/down)
\n- (a^2 = \frac{44}{3}) → (a = \sqrt{\frac{44}{3}} \approx 3.83)
\n- (b^2 = 33) → (b = \sqrt{33} \approx 5.74)
\n- (c^2 = a^2 + b^2 = \frac{44}{3} + 33 = \frac{44 + 99}{3} = \frac{143}{3}) → (c = \sqrt{\frac{143}{3}} \approx 6.89)", "The asymptotes are the lines:", "[
\ny - 3 = \pm \frac{a}{b}(x - 4) = \pm \sqrt{\frac{44/3}{33}}(x - 4) = \pm \sqrt{\frac{4}{3}}(x - 4)
\n]
\n[
\ny = \pm \frac{2}{\sqrt{3}}(x - 4) + 3
\n]", "---", "## Why This Equation Matters: Practical Applications", "Understanding and rewriting such equations is crucial in fields like:
\n- Physics: Modeling trajectories and orbital paths
\n- Engineering: Designing structures with hyperbolic shapes (e.g., cooling towers)
\n- Computer Graphics: Rendering realistic curves and surfaces
\n- Economics & Data Science: Analyzing constraints and optimization in hyperbolic spaces", "---", "## Key Takeaways for SEO:", "- This equation represents a hyperbola in standard form after completing the square.
\n- Centered at ((4, 3)), it opens vertically due to the positive (y^2) term.
\n- Key parameters include center ((h,k) = (4, 3)), (a^2 = \frac{44}{3}), (b^2 = 33), and foci along the vertical axis.
\n- Knowing how to convert such forms helps analyze conic sections in math, physics, and engineering contexts.
\n- Search terms: hyperbola equation conversion, complete the square conic sections, standard form hyperbola, center and axes of hyperbola.", "---", "## Final Thoughts", "Mastering equations like (4(x^2 - 8x) - 9(y^2 - 6y) = -115) empowers students and professionals alike to unlock deeper insights into conic geometry. Use spreadsheet tools or graphing software (like Desmos or GeoGebra) to visualize this hyperbola—confirms the symmetry, asymptotes, and curves beautifully.", "If you're learning algebra, calculus, or coordinate geometry, this equation is a foundational example of how algebraic manipulation reveals geometric truth—making it a must-study for anyone advancing in math.", "---", "Keywords: hyperbola equation, completing the square, conic sections, standard form hyperbola, center of hyperbola, asymptotes, algebraic geometry, conic curves, (4(x^2 - 8x) - 9(y^2 - 6y) = -115)", "Meta Description:
\nDeep dive into the equation (4(x^2 - 8x) - 9(y^2 - 6y) = -115). Learn how to rewrite it in standard form, identify its hyperbolic geometry, and explore real-world applications in physics and engineering. Perfect for students and professionals studying conic sections."]