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Inequalities with Quadratic Functions Solving inequality problems Solving inequality problems
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Quadratic inequalities …means “for what values of x is this quadratic above the x axis” ax 2 +bx+c>0 e.g. x 2 + x - 20 >0 …means “for what values of x is this quadratic below the x axis” ax 2 +bx+c<0 e.g. x 2 + x - 20 < 0
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Inequality Problems (1) The n th triangular number is given by: n(n+1) 2 A) Find the value of n that gives the first triangular number over 100 B) What is the first triangular number over 100 Pg 75 Q3 C) Find the value of n that gives the first triangular number over 1000. What is it?
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Inequality Problems (1) The n th triangular number is given by: n(n+1) 2 A) Find the value of n that gives the first triangular number over 100 Pg 75 Q3 n(n+1) 2 > 100 n(n+1)>200 n 2 + n > 200 n 2 + n - 200 > 0 If n 2 + n - 200 = 0 a = 1 b = 1 c = -200 n = -1 [(-1) 2 - (4 x 1 x -200)] 2 x 1 n = -1 [1 - -800] = -1 801 2 2 n = 13.65 or -14.65 13.65-14.65
![Inequality Problems (1) The n th triangular number is given by: n(n+1) 2 A) Find the value of n that gives the first triangular number over 100 Pg 75 Q3 n(n+1) 2 > 100 n(n+1)>200 n 2 + n > 200 n 2 + n > 0 If n 2 + n = 0 a = 1 b = 1 c = -200 n = -1 [(-1) 2 - (4 x 1 x -200)] 2 x 1 n = -1 [ ] = -1 n = or](http://images.slideplayer.com/15/4598396/slides/slide_4.jpg "Inequality Problems (1) The n th triangular number is given by: n(n+1) 2 A) Find the value of n that gives the first triangular number over 100 Pg 75 Q3 n(n+1) 2 > 100 n(n+1)>200 n 2 + n > 200 n 2 + n > 0 If n 2 + n = 0 a = 1 b = 1 c = -200 n = -1 [(-1) 2 - (4 x 1 x -200)] 2 x 1 n = -1 [ ] = -1 n = or")
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Inequality Problems (1) The n th triangular number is given by: n(n+1) 2 A) Find the value of n that gives the first triangular number over 100 Pg 75 Q3 n(n+1) 2 > 100 n(n+1)>200 n 2 + n > 200 n 2 + n - 200 > 0 13.65-14.65 n > 13.65 or n< -14.65 n =13.65 gives 100 n =14 will give the integer solution over 100 B) What is the first triangular number over 100 n(n+1) 2 14(14+1) 2 = 14 x 15/2 = 105
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Inequality Problems (1) The n th triangular number is given by: n(n+1) 2 C) Find the value of n that gives the first triangular number over 1000. What is it? Pg 75 Q3 n(n+1) 2 > 1000 n(n+1)>2000 n 2 + n > 2000 n 2 + n - 2000 > 0 If n 2 + n - 2000 = 0 a = 1 b = 1 c = -2000 n = -1 [(-1) 2 - (4 x 1 x -2000)] 2 x 1 n = -1 [1 - -8000] = -1 8001 2 2 n = 44.22 or -45.22 44.22-45.22 If n=45, number is 1035
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Inequality Problems (2) AQA 2002 Solve 2x 2 + 8x +7 = 0 Leaving answers as surds B) Hence solve 2x 2 + 8x +7 > 0 A) Solve 2x 2 + 8x +7 = 0 a = 2 b = 8 c = 7 x = -8 [(8) 2 - (4 x 2 x 7)] 2 x 2 x = -8 [64 - 56] = -8 8 4 4 x = -2 + 1 / 2 2 = -8 2 2 4 = -2 2 2 Or x = -2 - 1 / 2 2
![Inequality Problems (2) AQA 2002 Solve 2x 2 + 8x +7 = 0 Leaving answers as surds B) Hence solve 2x 2 + 8x +7 > 0 A) Solve 2x 2 + 8x +7 = 0 a = 2 b = 8 c = 7 x = -8 [(8) 2 - (4 x 2 x 7)] 2 x 2 x = -8 [ ] = -8 x = / 2 2 = -8 2 2 4 = -2 2 2 Or x = / 2 2](http://images.slideplayer.com/15/4598396/slides/slide_7.jpg "Inequality Problems (2) AQA 2002 Solve 2x 2 + 8x +7 = 0 Leaving answers as surds B) Hence solve 2x 2 + 8x +7 > 0 A) Solve 2x 2 + 8x +7 = 0 a = 2 b = 8 c = 7 x = -8 [(8) 2 - (4 x 2 x 7)] 2 x 2 x = -8 [ ] = -8 x = / 2 2 = -8 2 2 4 = -2 2 2 Or x = / 2 2")
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Inequality Problems (2) AQA 2002 Leaving answers as surds B) Hence solve 2x 2 + 8x +7 > 0 A) Solve 2x 2 + 8x +7 = 0 x = -2 + 1 / 2 2 Or x = -2 - 1 / 2 2 x > -2 + 1 / 2 2 Or x < -2 - 1 / 2 2
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