Question

If x= 1/2-√3 , prove that x³-2x²-7x+5=3

Answer 1

`x=(1)/(2)-\sqrt3\\\\x^3-2x^2-7x+5=3\\\\((1)/(2)-\sqrt3)^3-2((1)/(2)-\sqrt3)^2-7((1)/(2)-\sqrt3)+5\\\\=((1)/(2))^3-3\cdot((1)/(2))^2\cdto\sqrt3+3\cdot(1)/(2)\cdot(\sqrt3)^2-(\sqrt3)^3-...\\\\...-2[((1)/(2))^2-2\cdot(1)/(2)\cdot\sqrt3+(\sqrt3)^2]-(7)/(2)+7\sqrt3+5`


`=(1)/(8)-(3\sqrt3)/(4)+(9)/(2)-3\sqrt3-2((1)/(4)-\sqrt3+3)-(7)/(2)+7\sqrt3+5\\\\=(1)/(8)+(9)/(2)-(1)/(2)-6-(7)/(2)+5-(3)/(4)\sqrt3-3\sqrt3+2\sqrt3+7\sqrt3\\\\=(1)/(8)+(1)/(2)-1+5(1)/(4)\sqrt3=(1)/(8)-(1)/(2)+5(1)/(4)\sqrt3=(1)/(8)-(4)/(8)+5(1)/(4)\sqrt3\\\\=-(3)/(8)+5(1)/(4)\sqrt3\\eq3`