Question

Sa se descompuna in produs de factorii expresiile:
a)2x²-5x-3
b)-3x²+5x+2
c)x²+x-12
d)x²-(√2+√6)x+2√3
e)x²-√3-3x

Answer 1

a)Δ=25+24=49 =>√Δ=7
x₁=(5-7)/4=-1/2
x₂=(5+7)/4=3
=>2(x+1/2)(x-3) sau (2x+1)(x-3)

b)Δ=25+24=49 =>√Δ=7
x₁=(-5-7)/-6=2
x₂=(-5+7)/-6=-1/3
=>-3(x+1/3)(x-2) sau -(3x+1)(x-2) sau (3x+1)(2-x)

c)Δ=1+48=49 =>√Δ=7
x₁=(-1-7)/2=-4
x₂=(-1+7)/2=3
=>(x+4)(x-3)

d)Δ=(√2+√6)²-8√3
     =2+4√3+6-8√3
     =8-4√3
√Δ-formula radicalilor compusi :
$\sqrt{8- \sqrt{48} } = \sqrt{ \frac{8+ \sqrt{64-48} }{2} } - \sqrt{ \frac{8- \sqrt{64-48} }{2} }$
$= \sqrt{ \frac{8+4}{2} } - \sqrt{ \frac{8-4}{2} } = \sqrt{6} - \sqrt{2}$
x₁=(√2+√6-√6+√2)/2=2√2/2=√2
x₂=(√2+√6+√6-√2)/2=2√6/2=√6
=>(x-√2)(x-√6)

e)Δ=9+4√3
√Δ=√(9+√48)
De aici, nu se mai poate lucra cu formula radicalilor compusi, pentru ca √(81-48) nu da un rezultat pe care sa-l pot scoate, macar partial, de sub radical...