Exercice De Maths

[Huryu]

Voila des autres exos assez facile mais j'ai besoin d'explication sur la réalisation.

[Barbidoux]

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Exercice 1

Là rien à comprendre mais identitées remarquables à savoir par coeur....

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(2*a+b)^2=4*a^2+4*a*b+b^2

(a+3*b)^2=a^2+6*a*b+9*b^2

(2*a+3*b)^2=4*a^2+12*a*b+9*b^2

(a/3+b)^2=a^2/9+2*a*b/3+b^2

(a-2*b)^2=a^2-4*a*b+4*b^2

(3*a-b)^2=9*a^2-6*a*b+b^2

(2*a-4*b)^2=4*a^2-16*a*b+16*b^2

(a^2+b^2)^2=a^4+2*a^2*b^2+b^4

(2*a^2+3*b^2)^2=4*a^4+12*a^2*b^2+9*b^4

(a^2*b+3*b)^2=a^4*b^2+6*a^2*b^2+9*b^2

(2*a/3-b/4)^2=4*a^2/9-a*b/3+b^2/16

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Exercice 2

Là encore identitées remarquables à savoir par coeur....

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4*x^2+ ......*x+.....=( 2*x+5)^2

4*x^2+ 20*x+ 25=( 2*x+5)^2

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4 *x^2-9=(2*x+.....)*(.......-........)

4 *x^2-9=(2*x+ 3)*( 2*x- 3)

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x^2+14*x+........=( x+ 7)^2

x^2+14*x+ 49=( x+ 7)^2

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9*x^2-16=(3*x+......)(....-.....)

9-16=(3*x+ 4)( 3*x- 4)

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( 3*x/5+1/2)^2=....x^2+3*x/5+.......

(3*x/5+1/2)^2= (9/25)*)x^2+3*x/5+ 1/4)

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Exercice 3

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(6*x+3)-(x-4)*(2*x+1)=3*(2*x+1)-(x-4)*(2*x+1)=(2*x+1)*(3-(x-4))=(2*x+1)*(7-x)=0 ==> x=-1/2 et x=7

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3*(2*x-1)+(x+2)*(2-4*x)=3*(2*x-1)-(x+2)*(4*x-2)=3*(2*x-1)-2*(x+2)*(2*x-1)=(2*x-1)*(3-2*(x+2))=(2*x-1)*(-1-2*x) ==> x=1/2 et x=-1/2

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4*x^2-16-(2*x+3)*(x-2)=(2*x-4)*(2*x+4)-(2*x+3)*(x-2)=2*(x-2)*(2*x+4)-(2*x+3)*(x-2)=(x-2)*(2*(2*x+4)-(2*x+3))=(x-2)*(2*x+5) ==> x=2 et x=-5/2

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(2*x+5)*(2*x-4)-x^2+4=2*(2*x+5)*(x-2)-(x+2)*(x-2)=(x-2)*(2*(2*x+5)-(x+2))=(x-2)*(3*x+8) ==> x=2 et x=-8/3

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(x+1)*(x-3)+18-x^2=21-4*x ==> x=4/21

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(x-3)*(2*x-1)+12-4*x=(x-3)*(2*x-1)-4*(x-3)=(x-3)*(2*x-5) ==> x=3 et x=5/2

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Exercice 4

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f(x)=x^2-1=(x+1)*(x-1)

....................(-1)........................(1)...............

f(x)........(+)..(0)............(-).........(0)......(+)....

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f(x)=(x+1)^2-49=(x+1)^2-7^29=(x+8)*(x-6)

....................(-8)........................(6)...............

f(x)........(+)..(0)............(-).........(0)......(+)....

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f(x)=x^3+2*x+x=x*(x+1)^2

f(x)>0 pour x>0

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f(x)=(x^2-9)*(2*x+1)-(x-3)*(2*x+1)^2=(x-3)*(x+3)*(2*x+1)-(x-3)*(2*x+1)^2=(x-3)*(2*x+1)*((x+3)-(2*x+1))=(x-3)*(2*x+1)*(2-x)

...........................(-1/2)...............(2)......................(3)...............

(x-3)...........(-)..................(-)....................(-)..........(0)........(+)...

(2*x+1)......(-)........(0)......(+)..................(+)......................(+)...

(2-x)...........(+).................(+)......(0).........(-)..........(0)........(-)....

f(x)............(+).......(0).......(-).......(0).......(+)...........(0)........(-)....

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f(x)=(4*x^2-9)*(x+3)-(2*x+3)*(2*x^2+12*x+18)=(2*x+3)*(2*x-3)*(x+3)-2*(2*x+3)*(x^2+6*x+9)=(2*x+3)*(2*x-3)*(x+3)-2*(2*x+3)*(x+3)^2=-9*(2*x+3)*(x+3)=

....................(-3)........................(-3/2)...............

f(x)........(-)....(0)............(+).........(0)......(-)....

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Exercice 5

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soit x la partie brisée. la partie brisée et la partie non brisées forment un triangle rectangle dont les côtées de l’angle droit valant respectivement 10-x et 3 m et l’hypothénuse x m ==> x^2=(10-x)^2+3^3 ==> 109-20*x=0 ==> x=5,45 et la brisure se trouve à 4,55 m du sol

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Exercice 6

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(x-1)^2-4=x^2-2*x-3==(x-1-2)*(x-1+2)=(x-3)*(x+1)

....................(-1)........................(3)...............

f(x)........(+)..(0)............(-).........(0)......(+)....

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