Réponses

Meilleure réponse !
2014-05-17T00:28:16+02:00
Bonsoir,

1) démontrer que x²-6x-7=(x-3)²-16

(x - 3)² - 16 = (x² - 6x + 9) - 16
(x - 3)² - 16 = x² - 6x + 9 - 16
(x - 3)² - 16 = x² - 6x -7

2) determiner le signe de T(x)= x²-6x-7

T(x) = x² - 6x - 7
T(x) = (x - 3)² - 16
T(x) = (x - 3)² - 4²
T(x) = [(x - 3) - 4] [(x - 3) + 4]
T(x) = (x - 3 - 4)(x - 3 + 4)
T(x) = (x - 7)(x + 1)

Tableau de signes :

Racines : x - 7 = 0 ==> x = 7
               x + 1 = 0 ==> x = -1

\begin{array}{|c|ccccccc||}x&-\infty&&-1&&7&&+\infty\\ x-7&&-&-&-&0&+&\\ x+1&&-&0&+&+&+&\\ T(x)=(x-7)(x+1)&&+&0&-&0&+& \\\end{array}\\\\\\T(x)=x^2-6x-7\\\\\\x^2-6x-7>0\ \ si\ \ x\in]-\infty;-1[\ \cup\ ]7;+\infty[\\x^2-6x-7=0\ \ si\ \  x = -1\ \ ou\ \ x=7\\x^2-6x-7<0\ \ si\ \ x\in]-1;7[\end{matrix}\right.