Bonjour,
1) Développer A(x)
A(x)= (4x+1)²-(6x-11)²
A(x)= 16x²+8x+1-(36x²-132x+121)
A(x)= 16x²+8x+1-36x²+132x-121
A(x)= -20x²+140x-120
2) Factoriser A(x):
A(x)= (4x+1)²-(6x-11)²
A(x)= (4x+1-6x+11)(4x+1+6x-11)= 0
A(x)= (-2x+12)(10x-10)
A(x)= -20x(x-6)(x-1)
3) Montrer que A(x)= -20(x-7/2)²+125
On développe:
-20(x-7/2)(x-7/2)+125= -20(x²- 7x/2-7x/2+49/2) + 125
= -20(x²-14x/2+49/4)+125
= -20(4*x²/4-2*14x/4+49/4)+125
= -20(x² - 28x/4+ 49/4)+125
= -20(x²-7x+49/4) + 125
= -20x²+140x-245+125
A(x) = -20x²+140x -120
Donc
A(x)= (4x+1)²-(6x-11)²= -20(x-7/2)²+125
4) Résoudre :
A(x)= 0 ***** forme factorisée à calculer
(4x+1)²-(6x-11)²= 0
-20(x-6)(x-1) = 0
-x-6= 0 ou x-1= 0
x= 6 x= 1
S= { 1 ; 6)
A(x)= - 120 **** forme développée
-20x²+140x-120= -120
-20x²+140x-120+120=0
-20x²+140x= 0
-20x(x-7)= 0
-20x= 0 ou x-7= 0
x= 0 x= 7
S= { 0 ; 7 }
A(x)= 45 *** forme développée
-20x²+140x -120-45= 0
-20x²+140x -165= 0
Δ= b²-4ac= (140)²-4(-20)(-165)= 6 400 => √6 400= 80
x= (-140-80)/-40= 11/2
x2= (-140+80) / -40= 3/2
S= { 3/2 ; 11/2 }
(-10x+55)(2x-3)= 0
A(x)= -20x² forme développée.
-20x²+140x-120= -20x²
-20x²+140x-120+20x²= 0
140x-120= 0
20(7x-6)= 0
7x-6= 0
x= 6/7
S= { 6/7 }
