Quadratic Functions

Example 2The straight line y = 2 k + 1   intersects the curve y = x + k 2 x   at two distinct points.  Find the range of values of k.

Example 1Find the value of p for which 8 y = x + 2 p   is a tangent to the curve 2 y 2 = x + p  .

Nature of the Roots (Combination of Straight Line and the Curve)When you have a straight line and a curve, you can solve the equation of the straight line and the curve simultaneously and form a quadratic equation, ax2 +bx + c = 0. The discriminant, b 2 − 4 a c  gives information about the number of … 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

ExampleThe roots of 2 x 2 + 3 x − 1 = 0   are α and β.  Find the values of (a) ( α + 1 ) ( β + 1 ) (b) 1 α + 1 β (c) α 2 β + α β 2 (d) α β + β α [Clue: α 2 … 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