Question

13. Găsiți câte o ecuaţie de gradul al doilea care are rădăcinile indicate în fiecare caz: d) x₁ =1-√3, x₂ =1+√3; a) x₁ = 3, x₂ = -4; b) x₁ = -3, x₂ = 4; c) x₁ = 2, x₂ = -2; e) x₁ = a +b, x₂ = a-b, a, b apartine R; f) x₁ = 2-√2, x₂ = 2+√2.​

Answer 1

Răspuns:

Explicație pas cu pas:

d) x₁ =1-√3, x₂ =1+√3

(x-x₁)(x-x₂) = 0 <=>x²-xx₁-xx₂+x₁x₂

(x-1+√3)(x-1-√3) = x²-x(1-√3)-x(1+√3)+(1+√3)(1-√3) =

= x²-x+x√3-x-x√3+1-3 = x²-2x-2

x²-2x-2 = 0

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a) x₁ = 3, x₂ = -4 =>

(x-3)(x+4) = 0 => x²+4x-3x-12 = 0 =>

x²+x-12 = 0

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b) x₁ = -3, x₂ = 4 => (x+3)(x-4) = 0 =>

x²-x-12 = 0

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c) x₁ = 2, x₂ = -2 =>

(x-2)(x+2) = 0 =>

x²-4 = 0

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e) x₁ = a +b, x₂ = a-b

(x-a-b)(x-a+b) = 0 => x²-x(a+b)-x(a-b)+(a+b)(a-b) =

= x²-xa-xb-xa+xb+a²-b² = x²-2xa+a²-b²

x²-2ax+a²-b² = 0

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f) x₁ = 2-√2, x₂ = 2+√2

(x-2+√2)(x-2-√2) = 0 => x²-x(2-√2+2+√2)+(2-√2)(2+√2) = 0 =>

x²-4x+2 = 0