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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

Inequalities with Quadratic Functions

Solving inequality problems

quadratic inequalities
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

slide3

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?

slide4

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

slide5

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

slide6

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

slide7

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

slide8

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