Question

What loci are represented by the equation :
$(x^3-1)(y^2-4)=0.$

Answer 1

$(x^3-1)(y^2-4)=0$

Factorising the given equation

$(x-1)(x^2+x+1)(y+2)(y-2)=0(x-1)(y+2)(y-2)(x^2+x+\frac{1}{4}+\frac{3}{4})=0(x-1)(y+2)(y-2)({(x+\frac{1}{2})}^2+\frac{3}{4})=0$

Thus the equation represents three straight lines one parallel to $y$ axis and two parallel to $x$ axis and a parabola whose vertex is $(-\frac{1}{2},0)$.