Convert from expanded form

Learning Objectives Convert the general expanded form of a circle's equation to standard (center-radius) form. Convert the general expanded form of a parabola's equation to standard (vertex) form. Convert the general expanded form of an ellipse's equation to standard form. Convert the general expanded form of a hyperbola's equation to standard form. Identify the type of conic section from its general expanded equation. Fluently apply the 'completing the square' method to equations with both x² and y² terms. Extract key features (e.g., center, radius, vertices, foci) of a conic section after converting its equation from expanded form. Ever see a jumbled equation like 4x² - 9y² + 32x + 18y + 19 = 0 and wonder how it describes a perfect, geometri...

TermDefinitionExample Expanded Form (General Form)The form of a conic section equation where all terms are on one side, set equal to zero. The general form is Ax² + Cy² + Dx + Ey + F = 0 (for conics aligned with the axes).9x² + 16y² - 36x + 96y + 36 = 0 is the expanded form of an ellipse. Standard FormA specific format for a conic section's equation that transparently reveals its geometric properties like center, radius, vertices, and orientation.The standard form for a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Completing the SquareAn algebraic technique used to convert a quadratic expression of the form x² + bx into a perfect square trinomial, (x + b/2)², by adding the value (b/2)².To complete the square for x² - 10x, we add (-10/2)² = 25. T...

Standard Form of a Circle (x - h)² + (y - k)² = r² Used for circles. This form directly gives the center (h, k) and the radius r. Standard Forms of an Ellipse and Hyperbola \frac{(x - h)²}{a²} + \frac{(y - k)²}{b²} = 1 \quad \text{(Ellipse)} \quad \text{or} \quad \frac{(x - h)²}{a²} - \frac{(y - k)²}{b²} = 1 \quad \text{(Hyperbola)} Used for ellipses and hyperbolas. This form reveals the center (h, k) and the lengths of the axes, which are used to find vertices and foci. The key difference is the plus sign for an ellipse and the minus sign for a hyperbola. Standard Forms of a Parabola (x - h)² = 4p(y - k) \quad \text{(Vertical)} \quad \text{or} \quad (y - k)² = 4p(x - h) \quad \text{(Horizontal)} Used for parabolas. This form reveals the vertex (h, k), the direction...