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Block 2
Solving Quadratic Inequations
What is to be learned?
• The best tactic for solving quadratic
inequations
Solving Quadratic Equations
Solve x2
– 2x – 8 = 0
Factorise
(x – 4)(x + 2) = 0
x = 4 or x = -2
Big Nasty Formula
Big L
Trial and Error
Try x = 2 22
– 2(2) – 8 = -8 
Try x = 4  42
– 2(4) – 8 = 0 
22
Solving Quadratic Equations
Solve x2
– 2x – 8 = 0
Graphically
y = xy = x22
– 2x – 8– 2x – 8
-2-2 44-4-4
xx22
– 2x – 8 = 0– 2x – 8 = 0
x = -2 or 4x = -2 or 4
very easyvery easy ifif
you have a graphyou have a graph
Graphically is the way to go
Solve x2
+ 2x – 15 < 0
Need graph y = xy = x22
+ 2x – 15+ 2x – 15
Find RootsFind Roots FactoriseFactorise
(x + 5)(x – 3)(x + 5)(x – 3)
roots x=-5 or 3roots x=-5 or 3
Solving Quadratic Inequations
33
Solving Quadratic Inequations
Solve x2
+ 2x – 15 < 0
Roots x = -5 or 3
-5-5
xx22
+ 2x – 15 < 0+ 2x – 15 < 0
y = xy = x22
+ 2x – 15+ 2x – 15
y positivey positive
y negativey negative
Solving Quadratic Inequations
Solve x2
+ 2x – 15 < 0
Roots x = -5 or 3
-5-5
xx22
+ 2x – 15 < 0+ 2x – 15 < 0
xx is between -5 and 3is between -5 and 3y = xy = x22
+ 2x – 15+ 2x – 15
y positivey positive
y negativey negative
-5 < x < 3-5 < x < 3
33
Solving Quadratic Inequations
Best done by drawing a graph
For graph, need
For roots
rootsroots
FactoriseFactorise
Solve x2
+ x – 6 > 0
Need graph y = xy = x22
+ x – 6+ x – 6
Find RootsFind Roots FactoriseFactorise
(x + 3)(x – 2)(x + 3)(x – 2)
roots x= -3 or 2roots x= -3 or 2
22
Solve x2
+ x – 6 > 0
Roots x = -3 or 2
-3-3
xx22
+ x – 6 > 0+ x – 6 > 0
y = xy = x22
+ x – 6+ x – 6
y positivey positive
y negativey negative
22
Solve x2
+ x – 6 > 0
Roots x = -3 or 2
-3-3
xx22
+ x – 6 > 0+ x – 6 > 0
y = xy = x22
+ x – 6+ x – 6
y positivey positive
y negativey negative
x < -3x < -3 and x > 2and x > 2
Solve:
1. x2
– 5x + 6 < 0
2. x2
– 2x – 8 > 0
3. x2
– 16 < 0
4. x2
– 10x > 0
5. 10x – x2
> 0
6. x2
– 3x – 18 ≥ 0
7. 2x2
– 8x + 6 ≤ 0
8. x2
+ 8x + 16 < 0
9. x2
+ 8x + 16 ≤ 0
10. x2
+ 8x + 16 > 0 .
2 < x < 32 < x < 3
x > 4 or x < -2x > 4 or x < -2
Key QuestionKey Question
x > 10 or x < 0x > 10 or x < 0
0 < x < 100 < x < 10
xx ≥ 6 or x6 or x ≤ -2-2
11 ≤ xx ≤ 33
x2
– 16 < 0
Factorising (x – 4)(x + 4)
Roots x = 4 and x = -4
Key QuestionKey Question
44-4-4
-4 < x < 4-4 < x < 4

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Solving quadratic inequations

  • 2. What is to be learned? • The best tactic for solving quadratic inequations
  • 3. Solving Quadratic Equations Solve x2 – 2x – 8 = 0 Factorise (x – 4)(x + 2) = 0 x = 4 or x = -2 Big Nasty Formula Big L Trial and Error Try x = 2 22 – 2(2) – 8 = -8  Try x = 4  42 – 2(4) – 8 = 0 
  • 4. 22 Solving Quadratic Equations Solve x2 – 2x – 8 = 0 Graphically y = xy = x22 – 2x – 8– 2x – 8 -2-2 44-4-4 xx22 – 2x – 8 = 0– 2x – 8 = 0 x = -2 or 4x = -2 or 4 very easyvery easy ifif you have a graphyou have a graph
  • 5. Graphically is the way to go Solve x2 + 2x – 15 < 0 Need graph y = xy = x22 + 2x – 15+ 2x – 15 Find RootsFind Roots FactoriseFactorise (x + 5)(x – 3)(x + 5)(x – 3) roots x=-5 or 3roots x=-5 or 3 Solving Quadratic Inequations
  • 6. 33 Solving Quadratic Inequations Solve x2 + 2x – 15 < 0 Roots x = -5 or 3 -5-5 xx22 + 2x – 15 < 0+ 2x – 15 < 0 y = xy = x22 + 2x – 15+ 2x – 15 y positivey positive y negativey negative
  • 7. Solving Quadratic Inequations Solve x2 + 2x – 15 < 0 Roots x = -5 or 3 -5-5 xx22 + 2x – 15 < 0+ 2x – 15 < 0 xx is between -5 and 3is between -5 and 3y = xy = x22 + 2x – 15+ 2x – 15 y positivey positive y negativey negative -5 < x < 3-5 < x < 3 33
  • 8. Solving Quadratic Inequations Best done by drawing a graph For graph, need For roots rootsroots FactoriseFactorise
  • 9. Solve x2 + x – 6 > 0 Need graph y = xy = x22 + x – 6+ x – 6 Find RootsFind Roots FactoriseFactorise (x + 3)(x – 2)(x + 3)(x – 2) roots x= -3 or 2roots x= -3 or 2
  • 10. 22 Solve x2 + x – 6 > 0 Roots x = -3 or 2 -3-3 xx22 + x – 6 > 0+ x – 6 > 0 y = xy = x22 + x – 6+ x – 6 y positivey positive y negativey negative
  • 11. 22 Solve x2 + x – 6 > 0 Roots x = -3 or 2 -3-3 xx22 + x – 6 > 0+ x – 6 > 0 y = xy = x22 + x – 6+ x – 6 y positivey positive y negativey negative x < -3x < -3 and x > 2and x > 2
  • 12. Solve: 1. x2 – 5x + 6 < 0 2. x2 – 2x – 8 > 0 3. x2 – 16 < 0 4. x2 – 10x > 0 5. 10x – x2 > 0 6. x2 – 3x – 18 ≥ 0 7. 2x2 – 8x + 6 ≤ 0 8. x2 + 8x + 16 < 0 9. x2 + 8x + 16 ≤ 0 10. x2 + 8x + 16 > 0 . 2 < x < 32 < x < 3 x > 4 or x < -2x > 4 or x < -2 Key QuestionKey Question x > 10 or x < 0x > 10 or x < 0 0 < x < 100 < x < 10 xx ≥ 6 or x6 or x ≤ -2-2 11 ≤ xx ≤ 33
  • 13. x2 – 16 < 0 Factorising (x – 4)(x + 4) Roots x = 4 and x = -4 Key QuestionKey Question 44-4-4 -4 < x < 4-4 < x < 4