Complete the square

Learning Objectives Identify the value needed to create a perfect square trinomial. Solve quadratic equations of the form x² + bx + c = 0 by completing the square. Solve quadratic equations of the form ax² + bx + c = 0, where a ≠ 1, by completing the square. Convert a quadratic function from standard form (y = ax² + bx + c) to vertex form (y = a(x-h)² + k). Identify the vertex and axis of symmetry of a parabola from its vertex form. Explain how completing the square can be used to derive the quadratic formula. Ever wondered how to calculate the exact path of a basketball for a perfect swish? 🏀 Completing the square is a key tool for modeling that perfect arc! This tutorial will guide you through 'completing the square,' a powerful algebraic method. You will learn...

TermDefinitionExample Quadratic Equation (Standard Form)An equation that can be written in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.2x² - 5x + 3 = 0 is a quadratic equation in standard form. Perfect Square TrinomialA trinomial that is the result of squaring a binomial. It can be factored into the form (x + k)² or (x - k)².x² + 10x + 25 is a perfect square trinomial because it factors into (x + 5)². BinomialAn algebraic expression with two terms.(x + 5) is a binomial. Vertex FormA way of writing a quadratic function as y = a(x - h)² + k, where the point (h, k) is the vertex of the parabola.y = 2(x - 3)² + 1 is in vertex form. The vertex is at (3, 1). VertexThe highest or lowest point on a parabola. It represents the maximum or minimum value of the quadratic func...

The 'Completing the Square' Term For an expression x² + bx, the term to add to create a perfect square trinomial is (b/2)². This is the core of the method. Take the coefficient of the x-term (b), divide it by 2, and then square the result. This 'magic number' completes the square. Factoring the Perfect Square x² + bx + (b/2)² = (x + b/2)² Once you've added the correct term, the resulting trinomial always factors into a binomial squared. The term inside the parentheses is always x plus half of the original b-coefficient. The Square Root Property If X² = k, then X = ±√k This property is used in the final steps of solving the equation. When you take the square root of both sides of an equation, you must account for both the positive and negativ...