Inequalities with Quadratic Functions Solving inequality problems Quadratic

Inequalities with Quadratic Functions • Solving inequality problems

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

Inequality Problems (1) The nth 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 C) Find the value of n that gives the first triangular number over 1000. What is it? Pg 75 Q 3

Inequality Problems (1) The nth triangular number is given by: n(n+1) 2 A) Find the value of n that gives the first triangular number over 100 n(n+1) > 100 2 n(n+1)>200 n 2 + n > 200 n 2 + n - 200 > 0 -14. 65 13. 65 Pg 75 Q 3 a=1 b=1 c = -200 If n 2 + n - 200 = 0 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

Inequality Problems (1) The nth triangular number is given by: Pg 75 Q 3 n(n+1) 2 A) Find the value of n that gives the first triangular number over 100 n(n+1) > 100 2 n(n+1)>200 n 2 + n > 200 n 2 + n - 200 > 0 -14. 65 13. 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

Inequality Problems (1) The nth 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? n(n+1) > 1000 2 n(n+1)>2000 n 2 + n > 2000 n 2 + n - 2000 > 0 -45. 22 44. 22 Pg 75 Q 3 a=1 b=1 c = -2000 If n 2 + n - 2000 = 0 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 If n=45, number is 1035

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

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