HKDSE Mathematics Ronald Hui Tak Sun Secondary School

Missing Homework  Summer Holiday Homework 1  1, 10, 14  Summer Holiday Homework 2  1, 9, 14  SHW1-R1  9, October 2015 Ronald HUI

Missing Homework  SHW2-A1  10  SHW2-B1  9  SHW2-C1  8, 9, 10, 12, 13, 14, 20  RE2  9 22 October 2015 Ronald HUI

Missing Homework  SHW2-R1  I got 8 only!  SHW2-P1  I got 4 only!!!  SHW3-01, 3-A1, 3-B1  Today!!!! 22 October 2015 Ronald HUI

Summary on “AND” 22 October 2015 Ronald HUI

Summary on “OR” 22 October 2015 Ronald HUI

Book 5A Chapter 3 Solving Quadratic Inequalities in One Unknown by the Graphical Method

If an inequality contains only one unknown and the highest degree of the terms is 2, the inequality is called a quadratic inequality in one unknown. Example: (i) x 2 + 3x – 10 > 0 (ii) x 2 – 2x + 1  0 (iv) –2x 2 + x + 1  0 (iii) x 2 + 3x – 10 < 0

We have to find all values of x satisfying the quadratic inequality. How can we solve this kind of quadratic inequalities?

Let’s see how to solve x 2 + 2x – 3 > 0 by the graphical method.

Solve the quadratic inequality x 2 + 2x – 3 > 0 graphically. Consider the graph of the quadratic function y = x 2 + 2x – 3. Points above the x-axis Points below the x-axis y > 0 y < 0

Solve the quadratic inequality x 2 + 2x – 3 > 0 graphically. Consider the graph of the quadratic function y = x 2 + 2x – 3. For x < –3: The graph is above the x-axis. For –3 < x < 1: The graph is below the x-axis. For x > 1: The graph is above the x-axis. y > 0 y < 0 y > 0

For y > 0 (i.e. x 2 + 2x – 3 > 0), ∴ The solutions of x 2 + 2x – 3 > 0 are x 1. For x < –3: y > 0 For –3 < x < 1: y < 0 For x > 1: y > 0 Solve the quadratic inequality x 2 + 2x – 3 > 0 graphically. Consider the graph of the quadratic function y = x 2 + 2x – 3. For y > 0 (i.e. x 2 + 2x – 3 > 0), the range of values of x are x 1.

Solve the quadratic inequality x 2 + 2x – 3 > 0 graphically. Consider the graph of the quadratic function y = x 2 + 2x – 3. Using the above graph, we can also find the solutions of x 2 + 2x – 3  0. For x 0 For –3 < x < 1: y < 0 For x > 1: y > 0

Solve the quadratic inequality x 2 + 2x – 3  0 graphically. For x  –3: The graph is on or above the x-axis. y  0 For –3  x  1: The graph is on or below the x-axis. y  0 For x  1: The graph is on or above the x-axis. y  0

the range of values of x are –3  x  1. Solve the quadratic inequality x 2 + 2x – 3  0 graphically. Note: In fact, the inequalities x 2 + 2x – 3  0 and x 2 + 2x – 3 < 0 can also be solved using the graph above. ∴ The solutions of x 2 + 2x – 3  0 are –3  x  1. For y  0 (i.e. x 2 + 2x – 3  0), For x  –3: y  0 For –3  x  1: y  0 For x  1: y  0

Let’s see how we can solve the quadratic inequality x 2 + 3x – 10 < 0 without any given graph.

Consider the corresponding quadratic function Solve the quadratic inequality x 2 + 3x – 10 < 0 graphically. Step 1 Write down the corresponding quadratic function y = ax 2 + bx + c. y = x 2 + 3x – 10.

Step 2 Find the x-intercept(s) and the direction of opening of the graph of the quadratic function. Then, sketch the graph. Solve the quadratic inequality x 2 + 3x – 10 < 0 graphically. Sketch the graph of y = x 2 + 3x – 10. x 2 + 3x – 10 = 0 (x + 5)(x – 2) = 0 x = –5 or x = 2 When y = 0,  ∴ The x-intercepts of the graph are –5 and 2.

Step 2 Find the x-intercept(s) and the direction of opening of the graph of the quadratic function. Then, sketch the graph. Solve the quadratic inequality x 2 + 3x – 10 < 0 graphically. Sketch the graph of y = x 2 + 3x – 10.  ∵ The coefficient of x 2 is 1 (> 0). ∴ The graph opens upwards. y = x 2 + 3x – 10

Read the solutions of the quadratic inequality from the graph. Step 3 y = x 2 + 3x – 10 For y < 0 (i.e. x 2 + 3x – 10 < 0), the corresponding part of the graph is below the x-axis. Solve the quadratic inequality x 2 + 3x – 10 < 0 graphically.

y = x 2 + 3x – 10 ∵ The graph of y = x 2 + 3x – 10 is below the x-axis when –5 < x < 2. ∴ The solutions of x 2 + 3x – 10 < 0 are –5 < x < 2. Solve the quadratic inequality x 2 + 3x – 10 < 0 graphically. Read the solutions of the quadratic inequality from the graph. Step 3

Follow-up question Solve the quadratic inequality –2x 2 + x + 1  0 graphically. Consider the corresponding quadratic function y = –2x 2 + x + 1. When y = 0, –2x 2 + x + 1 = 0 2x 2 – x – 1 = 0 (2x + 1)(x – 1) = x = – or x = 1 ∴ The x-intercepts of the graph of y = –2x 2 + x + 1 are – and

∵ The coefficient of x 2 is –2 (< 0). ∴ The graph of y = –2x 2 + x + 1 opens downwards. Sketch the graph of y = –2x 2 + x + 1: From the graph, the solutions of –2x 2 + x + 1  0 are x  – or x 