Convert between standard and expanded form

Learning Objectives Convert the general form of a circle's equation to its standard form by completing the square. Identify the key features of a conic section (e.g., center, radius, vertex) from its standard form equation. Convert the standard form of an ellipse or hyperbola's equation to its general (expanded) form by expanding binomials. Transform the general form of a parabola's equation to its standard form to find its vertex and orientation. Apply the process of completing the square to equations with both x² and y² terms, including those with leading coefficients. Recognize the type of conic section from its general form equation based on its x² and y² coefficients. Ever wonder how a GPS pinpoints your location or how a satellite dish focuses a signal?...

TermDefinitionExample Standard Form (of a Conic Section)The form of a conic section's equation that clearly reveals its geometric properties, such as its center, vertices, and radius.The standard form of a circle is (x-h)² + (y-k)² = r², where the center is (h, k) and the radius is r. Expanded Form (General Form)The form of a conic section's equation where all terms are multiplied out and typically set equal to zero, written as Ax² + By² + Cx + Dy + E = 0.The expanded form of a circle could be x² + y² - 6x + 2y - 6 = 0. 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, you add (10/2)² = 25 to get x² + 10x + 25, which fa...

Completing the Square For an expression x² + bx, add (b/2)² to create a perfect square trinomial: x² + bx + (b/2)² = (x + b/2)². This is the primary method for converting from expanded form to standard form. Remember to add the same value to both sides of the equation to keep it balanced. Standard Form of a Circle (x-h)² + (y-k)² = r² Represents a circle with center (h, k) and radius r. This is the target form when converting a circle's equation from general form. Standard Form of a Horizontal Parabola (y-k)² = 4p(x-h) Represents a parabola that opens left or right with its vertex at (h, k). This is the target form for parabolas with a y² term. Perfect Square Trinomial Expansion (a ± b)² = a² ± 2ab + b² This is the formula used to convert from standard f...