Question

select the equations of all lines through the orgin that are tangent to the curve y=x^3-8x^2-7x

Answer 1

y'=3x^2-16x-7

y=mx+b

y=(3x^2-16x-7)x+b on the line.

b=o, as it goes through origin.

y=x^3-8x^2-7x on the curve,

so, where are x,y the same on the line and curve.
Y line=ycurve
3x^3-16x^2-7x=x^3-8x^2-7x
2x^3-8x=0
2x(x^2-4)=0
x=0, x=+-2
so you have these points for x, figure y at those points and you have it.