4.
[tex]a.\\\\
x^2-2x-1=0\\
\Delta=b^2-4*a*c\\
\Delta=(-2)^2-4*1*(-1)\\
\Delta=4+4\\
\Delta=8\\\\
\bold{ \sqrt{\Delta} = \sqrt{8}=2 \sqrt{2}}\\ \\
x_1= \frac{-b+\sqrt{\Delta} }{2*a} = \frac{-(-2)+2 \sqrt{2} }{2} = \frac{2+2 \sqrt{2} }{2}= \frac{2*(1+ \sqrt{2} }{2}\bold{=1+ \sqrt{2} }\\\\
x_2= \frac{-b-\sqrt{\Delta} }{2*a} = \frac{-(-2)-2 \sqrt{2} }{2} = \frac{2-2 \sqrt{2} }{2}= \frac{2*(1-\sqrt{2} }{2}\bold{=1- \sqrt{2} }\\\\\\
b~,~c~si~d~sunt~foarte asemanatoare~ cu~ punctul ~a. \uparrow~ \uparrow~ \uparrow\\\\
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[tex]e.\\x^2-4=0\\
(x-2)(x+2)=0~\Longrightarrow~ \left {{(x-2)=0~\Longrightarrow~\bold{x=2}} \atop {(x+2)=0 ~\Longrightarrow~\bold{x=-2}} \right. [/tex]
[tex]f.
\\
2x^2+3x=0~\Longrightarrow~x*(2x+3)=0~\Longrightarrow~ \left {{x=0} \atop {2x+3=0 \Longrightarrow2x=-3 \Longrightarrow~x= -\frac{3}{2} } \right. [/tex]
[tex]5.\\
a.\\\\x^2-1=x-3\\
x^2-x-1+3=0\\
x^2-x+2=0\\
\Delta=b^2-4*a*c=(-1)^2-4*1*2=1-8=-7\\\Longrightarrow~Nu ~esista~solutii!!!\\\\
b.\\
2(x+1)-x(x+1)=0\\
2x+2-x^2-1=0\\
-x^2+2x+1=0\\
\Delta=b^2-4*a*c=2^2-4*(-1)*1=4+4=8\\
\sqrt{\Delta}=\sqrt{8}= 2\sqrt{2} \\\\
x_1= \frac{-b+\sqrt{\Delta}}{2*a}= \frac{-2+2 \sqrt{2}}{-2}= \frac{-2*(1- \sqrt{2}}{-2}=\bold{1- \sqrt{2}} \\\\
x_2= \frac{-b-\sqrt{\Delta}}{2*a}= \frac{-2-2 \sqrt{2}}{-2}= \frac{-2*(1+ \sqrt{2}}{-2}=\bold{1+ \sqrt{2}} \\\\
[/tex]
[tex]c.\\\\
x^2-6x+9=4\\
x^2-6x+9-4=0\\
x^2-6x+5=0\\
\Delta=b^2-4*a*c=36-4*1*5=36-20=16\\
\sqrt{\Delta} = \sqrt{16}=4\\\\
x_1= \frac{-b+\sqrt{\Delta}}{2*a}= \frac{-(-6)+4}{2} = \frac{6+4}{2}= \frac{10}{2} \bold{=5}\\\\
x_2= \frac{-b-\sqrt{\Delta}}{2*a}= \frac{-(-6)-4}{2} = \frac{6-4}{2}= \frac{2}{2} \bold{=1}\\\\\\\\
[/tex]
[tex]d. \\
\\\frac{x^2-1}{2} = \frac{x+1}{3}\\
2*(x+1)=3*(x^2-1)\\
2x+1=2x^2-3\\
-2x^2+2x+1+3=0\\
-2x^2+2x+4=0\\
\Delta=b^2-4*a*c=4+32=36\\
\sqrt{\Delta}= \sqrt{36}=6 \\\\
x_1= \frac{-b+\sqrt{\Delta}}{2*a}= \frac{-2+6}{2*(-2)} = \frac{4}{-4} =\bold{-1}\\
x_2=\frac{-b-\sqrt{\Delta}}{2*a}= \frac{-2-6}{2*(-2)} = \frac{-8}{-4} =\bold{2}\\\\
e.\\
1-(x-2)^2=0\\
1-x^2+2x-4=0\\
-x^2+2x-3=0\\
\Delta=b^2-4*a*c=4-12=-8\\
=\ \textgreater \ Ecuatia~nu~are~solutie~deoarece~\bold{\Delta\ \textless \ 0~}\\[/tex]
