Explications étape par étape :

1.) a) C(x) = (9x² - 25) - (6x + 10)(2x - 1)

= 9x² - 25 - 12x + 6x + 10 - 2x² + 6x

= -2x² + x + 15

b) C(x) = -15

-2x² + x + 15 = -15

-2x² + x - 30 = 0

(2x - 10)(-x + 3) = 0

x = 5 ou x = -3/2

2.) a) D(x) = (r + 6x + 9) + 4x(x + 3) - 9

= r + 6x + 9 + 4x² + 12x - 9

= 4x² + 18x + (r - 9)

b) D(x) = 4x

4x² + 18x + (r - 9) = 4x

4x² - 14x + (r - 9) = 0

2x(2x - 7) + (r - 9) = 0

3.) a) C(x) = (-2x + 1)(x - 15)

b) C(x) = 0

(-2x + 1)(x - 15) = 0

x = 1/2 ou x = 15

4.) a) D(x) = (2x - 7)(2x + (r - 9)/2)

b) D(x) = 0

(2x - 7)(2x + (r - 9)/2) = 0

x = 7/2 ou x = (9 - r)/4

5.) C(x) = D(x)

-2x² + x + 15 = 4x² + 18x + (r - 9)

-6x² - 17x + (r - 24) = 0

6.) C(0) = (-2)(0)² + 0 + 15 = 15

C(-1) = (-2)(-1)² + (-1) + 15 = 16

D(-2) = (2)(-2) + (r - 9) = r - 6

D = r - 6

7.) a) F(x) existe pour toutes les valeurs de x sauf x = 7/2 et x = (9 - r)/4.

b) F(x) = (-2x + 1)(x - 15)/(2x - 7)(2x + (r - 9)/2)

c) F(x) = 1

(-2x + 1)(x - 15)/(2x - 7)(2x + (r - 9)/2) = 1

d) F(-2) = (-2)(-2) + 1)/(2(-2) - 7)(2(-2) + (r - 9)/2) = (5 - 7(r - 9)/4)/(2(r - 9))

F(-2) = (-2(r - 9) + 4)/(2(r - 9)) = (-r + 13)/(2r - 18)