Question

Rezolvati in R ecuatiile:

Answer 1

 

$\displaystyle\\9)\\a)\\\\x^2-\left(\sqrt{2}+\sqrt{3}\right)x+\sqrt{6}=0\\\\x^2-\left(\sqrt{2}+\sqrt{3}\right)x+\sqrt{2\cdot3}=0\\\\x^2-\left(\sqrt{2}+\sqrt{3}\right)x+\sqrt{2}\cdot\sqrt{3}=0\\\\x^2-Sx+P=0\\\\x_1+x_2=\sqrt{2}+\sqrt{3}\\\\x_1\cdot x_2=\sqrt{2}\cdot\sqrt{3}\\\\\implies~\boxed{x_1=\sqrt{2}~~\text{si}~~x_2=\sqrt{3}}\\\\\\b)\\\\x^2-x+\frac{1}{4}=0\\\\ x^2-2\cdot\frac{1}{2}x+\left(\frac{1}{2}\right)^{\b2}=0\\\\\left(x-\frac{1}{2}\right)^{\b2}=0\\\\\implies\boxed{x_1=x_2=\frac{1}{2}}$

$\displaystyle\\c)\\\\x^2-2x+9=0\\\\\Delta=b^2-4ac=(-2)^2-4\cdot1\cdot9=4-36=-32\\\\\Delta<0\\\\\boxed{\text{Ecuatia nu are solutii in R.}}$