Question

Determine whether the following system possesses: (a) no solution, (b) a unique solution, or (c) many (how many?) solutions. (a) Xi + 3X^2 + X^3 = 0(b) 5x^2 - 6x^3 + x^4 = 0 (c) x - 2x^2 + 4x^3 = 2.

Answer 1

a) Have 3 solutions (0; -0.38...; -2.618...)

b) Have 3 solutions (0; 5; 1)

c) Have 1 real solution and 2 complex

Explanation:

a) x^3+3x^2+x=x(x^2+3x+1)=0, so if x=0 the equation is zero too, and if x^2+3x+2=0 the eq is zero again. To find the roots of x^2+3x+2, I have to use the quadratic formula,
`(-b+-\sqrt{b^(2)-4ac } ) /2a`, with a=1, b=3 and c=1 the solutions are
`(-3-√(5) )/(2)` and
`(-3+√(5) )/(2)`

b) x^4-6x^3+5x^2=x^2(x^2-6x+5), so if x^2=0 the equations 0, then x=0 is a double root, if I want to know the value of x for x^2-6x+5=0, then I use the quadratic formula again with a=1, b=-6 and c=5, and the solutions are 5 and 1

c) I couldn't calculate this with the analytical methods, so I had to do an aproximation and the unique real solution was approximately 0.87