Maximum and Minimum Point A quadratic functions f ( x ) = a x 2 + b x + c can be expressed in the form f ( x ) = a ( x + p ) 2 + q by the method of completing the square. The minimum/maximum point can be determined from the … Read more

2.8 The Graph of Quadratic Function The graph of quadratic function is parabola. When the coefficient of x2 is positive the graph is a parabola with U shape. When the coefficient of x2 is negative the graph is a parabola with ∩ shape. Axis of Symmetry The axis of symmetry is a vertical line passing through the maximum or … Read more

2.1 General Form of Quadratic Function General form of a quadratic function is f ( x ) = a x 2 + b x + c where a, b, and c are constants and a ≠ 0, and x as a variable. Example: Determine which of the following is a quadratic function.f ( x ) … Read more

The Discriminant The expression b 2 − 4 a c in the general formula is called the discriminant of the equation, as it determines the type of roots that the equation has. Example Determine the nature of the roots of the following equations. a. 5 x 2 − 7 x + 3 = 0 b. … Read more

2.4.1 Finding the Sum of Roots (SoR) and Product of Roots (PoR) Example Find the sums and products of the roots of the following equations. a. x 2 + 7 x − 3 = 0 b. x ( x − 1 ) = 5 ( 1 − x ) Answer: (a) x 2 + 7 … Read more

2.4 Forming Quadratic Equations from Given Roots Let α and β be the roots of the equation ax2 + bx + c = 0, this means Example: Find the quadratic equation from the respective roots below: (a) 3, −1 (b) −2, ¼ (c) ⅔, ¼ (d) 3m, −2m Solution: Click on the image below to … Read more

The quadratic equation a x 2 + b x + c = 0 can be solved by using the quadratic formula − b ± b 2 − 4 a c 2 a Example  Use the quadratic formula to find the solutions of the following equations. a. x 2 + 5 x − 24 = 0 … Read more

2.3.2 Solving Quadratic Equations – Completing the Square(A) The Perfect Square 1. The expression x2 + 2x + 1 can be written in the form (x + 1)2, it is called a “perfect square”.   2. If the algebraic expression on the left hand side of the quadratic equation is a perfect, the roots can be easily obtained … Read more

2.3.1 Solving Quadratic Equations – Factorisation 1. If a quadratic equation can be factorised into a product of two factors such that (x – p)(x – q) = 0 Hence  x – p = 0   or  x – q = 0    x = p   or x = q p and q  are the roots of the equation. Notes … Read more

Solving Quadratic Equations To solve a quadratic equation means to find all the roots of the quadratic equations. Example: Find the roots of the quadratic equations a. x 2 = 9 b. 2 x 2 − 98 = 0 Answer:  (a) x 2 = 9 x = ± 9 x = ± 3 (b) 2 … Read more