Question

Scrieti expresiile ca diferente de patrate si apoi descompuneti in factori:
a)x²-20x+96=
b)x²-8x-9=

Answer 1

x²  - 2·10x + 100 - 4 = ( x - 10)²  - 2² = ( x -10 + 2)( x -10 -2) 
                                 = ( x - 8)(x -12)

x² - 2 ·4x  + 16  -25  = ( x - 4)² -  5² = ( x -4 - 5)(x - 4 +5) =
                                     = ( x -9)(x+1)

Answer 2

[tex]a)x^{2} -20x+96=0\\ \Delta=b^2-4ac=(-20)^2-4*1*96=400-384=16\\x_1=\frac{-b-\sqrt{\Delta}}{2a}=\frac{20-4}{2}=\frac{16}{2}=8 \\ x_2=\\ ax^2+bx+c=a(x-x_1)(x-x_2)\\ x^{2} -20x+96=(x-8)(x-12)\\ x^{2} -20x+96=x^2-2x*10+100-100+96\\ x^{2} -20x+96=(x-10)^2-4\\ x^2-20x+96=(x-10)^2-2^2[/tex]
$b) x^{2} -8x-9=0\\ \Delta=b^2-4ac=(-8)^2-4*1*(-9)=64+36=100\\ x_1=\frac{-b-\sqrt{\Delta}}{2a}=\frac{8-10}{2}=\frac{-2}{2}=-1\\ x_2=\frac{-b+\sqrt{\Delta}}{2a}=\frac{8+10}{2}=\frac{18}{2}=9\\ x^{2} -8x-9=(x+1)(x-9) x^{2} -8x-9=x^2-2x*4-16+16-9\\ x^{2} -8x-9=(x-4)^2-7\\ x^{2} -8x-9=(x-4)^2-(\sqrt{7})^2$