Question

Rezolvati ecuatiile
x^2+x=6,x€Z
x^2-2x-3=0,x€R
4x^2-4x+1=0,x€Q
2x^2+X+1=0,x€R
Multumesc

Answer 1

1.
x²+x=6
x²+x-6=0
x²+3x-2x-6=0
x(x+3)-2(x+3)=0
(x+3)(x-2)=0
I. x+3=0 ⇒ x=-3
II. x-2=0 ⇒ x=2

S={-3,2}

2.

x²-2x-3=0
x²+x-3x-3=0
x(x+1)-3(x+1)=0
(x+1)(x-3)=0
I. x+1=0 ⇒ x=-1
II. x-3=0 ⇒ x=3

S={-1,3}

3.
4x²-4x+1=0
(2x)²-2·2x·1+1²=0
(2x-1)²=0
2x-1=0
2x=1 ⇒ x=1/2

S={1/2}

4.
2x²+x-1=0
2x²-x+2x-1=0
x(2x-1)+(2x-1)=0
(2x-1)(x+1)=0
I. 2x-1=0 ⇒ 2x=1 ⇒ x=1/2
II. x+1=0 ⇒ x=-1

S={-1,1/2}

Answer 2

x² + x = 6,  x∈Z

x² + x + 6 ⇒ x(x + 1) = 6 = -3·(-2) = 2·3 ⇒  x₁ = - 3,   x₂ = 2 .

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x² - 2x - 3 = 0,  x∈R

x² - 2x - 3 = 0 ⇒ x² +x -3x - 3 = 0 ⇒ x(x + 1) -3(x + 1) = 0 ⇒

⇒ (x + 1)(x - 3) = 0  ⇒ x₁ = - 1,   x₂ = 3 

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  4x² - 4x + 1= 0 ⇒ (2x - 1)² = 0 ⇒ 2x - 1 = 0 ⇒ x = 1/2

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2x² + x +1 = 0,  x€R


2x² + x +1 = 0 ⇔ x² + x² +x + 1/4  +3/4 = 0 ⇔ x² + 3/4 + (x + 1/2)² = 0

Membrul stâng al ecuației este strict mai mare decât 0, deci, ecuația

 nu admite soluții reale.