Question

Se considera x₁,x₂ solutiile ecuatiei x²-2x-2=0. Numarul x₁² + x₂² este egal cu?

Answer 1

Mai întâi afli soluțiile ecuției prin delta

x²-2x-2 = 0

Δ = 4+8 = 12

x₁ = (2+2√3)/2 = 2(1+√3)/2 = 1+√3

x₂ = (2-2√3)/2 = 2(1-√3)/2 = 1-√3

Calculezi suma dată

x₁² + x₂² = (1+√3)² + (1-√3)² = 1+2√3+3 + 1-2√3+3 = 8

Answer 2

$x^2-2x-2=0\\\\a=1,~b=-2,~c=-2\\\\\Delta=b^2-4ac=(-2)^2-4\cdot1\cdot(-2)=4+8=12 > 0\\\\x_1=\dfrac{-b-\sqrt{\Delta} }{2a}=\dfrac{-(-2)-\sqrt{12} }{2\cdot1} =\dfrac{2-2\sqrt{3} }{2}=\dfrac{\not2(1-\sqrt{3} )}{\not2} =1-\sqrt{3} \\\\x_2=\dfrac{-b+\sqrt{\Delta} }{2a}=\dfrac{-(-2)+\sqrt{12} }{2\cdot1} =\dfrac{2+2\sqrt{3} }{2}=\dfrac{\not2(1+\sqrt{3} )}{\not2} =1+\sqrt{3}$

$x_1^2+x_2^2=(1-\sqrt{3})^2+(1+\sqrt{3})^2=\\\\=1^2-2\cdot1\cdot\sqrt{3}+\Big(\sqrt{3}\Big)^2 +1^2+2\cdot1\cdot\sqrt{3}+\Big(\sqrt{3}\Big)^2=\\\\=1-2\sqrt{3}+3+1+2\sqrt{3}+3=8 ~~~~~~~~\boxed{C}$