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Inequalities with Quadratic Functions

Inequalities with Quadratic Functions. Solving inequality problems. 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.

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Inequalities with Quadratic Functions

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  1. Inequalities with Quadratic Functions Solving inequality problems

  2. Quadratic inequalities …means “for what values of x is this quadratic above the x axis” ax2+bx+c>0 e.g. x2+ x - 20 >0 …means “for what values of x is this quadratic below the x axis” ax2+bx+c<0 e.g. x2+ x - 20 < 0

  3. Pg 75 Q3 Inequality Problems (1) n(n+1) 2 The nth triangular number is given by: A) Find the value of n that gives the first triangular number over 100 B) What is the first triangular number over 100 C) Find the value of n that gives the first triangular number over 1000. What is it?

  4. Pg 75 Q3 n(n+1) 2 > 100 n = -1 [(-1)2 - (4 x 1 x -200)] 2 x 1 n = -1 [1 - -800] = -1 801 2 2 -14.65 13.65 Inequality Problems (1) n(n+1) 2 The nth triangular number is given by: A) Find the value of n that gives the first triangular number over 100 a = 1 b = 1 c = -200 n(n+1)>200 n2 + n > 200 n2 + n - 200 > 0 If n2 + n - 200 = 0 n = 13.65 or -14.65

  5. Pg 75 Q3 n(n+1) 2 > 100 n(n+1) 2 14(14+1) 2 -14.65 13.65 Inequality Problems (1) n(n+1) 2 The nth triangular number is given by: A) Find the value of n that gives the first triangular number over 100 n > 13.65 or n< -14.65 n =13.65 gives 100 n(n+1)>200 n2 + n > 200 n2 + n - 200 > 0 n =14 will give the integer solution over 100 B) What is the first triangular number over 100 = 14 x 15/2 = 105

  6. Pg 75 Q3 n(n+1) 2 > 1000 n = -1 [(-1)2 - (4 x 1 x -2000)] 2 x 1 n = -1 [1 - -8000] = -1 8001 2 2 -45.22 44.22 Inequality Problems (1) n(n+1) 2 The nth triangular number is given by: C) Find the value of n that gives the first triangular number over 1000. What is it? a = 1 b = 1 c = -2000 n(n+1)>2000 n2 + n > 2000 n2 + n - 2000 > 0 If n2 + n - 2000 = 0 n = 44.22 or -45.22 If n=45, number is 1035

  7. x = -8 [(8)2 - (4 x 2 x 7)] 2 x 2 = -2 2 2 Inequality Problems (2) A) Solve 2x2+ 8x +7 = 0 AQA 2002 Leaving answers as surds B) Hence solve 2x2+ 8x +7 > 0 Solve 2x2 + 8x +7 = 0 a = 2 b = 8 c = 7 x = -8 [64 - 56] = -8 8 4 4 = -8 22 4 x = -2 + 1/22 Or x = -2 - 1/22

  8. Inequality Problems (2) A) Solve 2x2+ 8x +7 = 0 AQA 2002 Leaving answers as surds B) Hence solve 2x2+ 8x +7 > 0 x = -2 + 1/22 Or x = -2 - 1/22 Or x < -2 - 1/22 x > -2 + 1/22

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