Cubic equation help need the three answers?

Question

I learned how to solve cubics but i cant remember and i need someone to solve this to get the thing that looks like (x-a)(x²-a-b) looking thing but i dont remember how to that. So please show me how to get the three answers for the following equation:

4x³-110x²+750x-1225=0

Answer 1

Unfortunately, the roots of this equation are not simple,

so it's either Newton's iteration or the following :

4x^3 - 110x^2 + 750x - 1225 = 0

Divide through by 4 :

x^3 - 27.5x^2 + 187.5x - 306.25 = 0

For the general cubic, x^3+ ax^2 + bx + c, we have

the equation with a = -27.5,  b = 187.5 and c = -306.25.

Calculate the following:

Q = (3b - a^2) / 9 = -21.527777778

R = (9ab - 27c - 2a^3) / 54 = 64.0046296296

Discriminant: D = Q^3 + R^2 = -5880.3530092593

If D > 0, there is 1 real root and 2 complex conjugate roots.

If D = 0, all roots are real and at least 2 are equal.

If D < 0, all roots are real and unequal.

Because D < 0, all roots are real and unequal.

Calculate the following:

cos(t) = R / sqrt(-Q^3)

= 0.6407853679

Therefore, t = 50.1495926363º

t / 3 = 16.7165308788º

cos(t / 3) = 0.9577395462

cos(t / 3 + 120º) = -0.7279705985

cos(t / 3 + 240º) = -0.2297689477

Solutions are given by:

x1 = 2*sqrt(-Q)*cos(t / 3) - a/3

x2 = 2*sqrt(-Q)*cos(t / 3 + 120º) - a/3

x3 = 2*sqrt(-Q)*cos(t / 3 + 240º) - a/3

That is,

x1 = 18.0541135239

x2 = 2.4113854076

x3 = 7.0345010686

So, 4x^3 - 110x^2 + 750x - 1225 = 0

is the same as :

4(x - 2.4114)(x - 7.0345)(x - 18.0541) = 0

Answer 2

In order to make the numbers more manageable, we'll start by substituting x = 5y to get

500 y^3 - 2650 y^2 + 3750 y -1225 = 0.  Divide through by 25:to get 20 y^3 -110 y^2 + 150 y - 49 = 0.  Some experimenting shows that this has three real roots: one between 0 and 1, one between 1 and 2, and one between 3 and 4.  These could be refined by Newton-Raphson iteration.  There is no obvious factorization of either the original or the transformed equation.

Answer 3

This is not a perfect cubic....Have tried x-a with a=5,7,25, etc all multiples of the last term. None come out even.

4x³-110x²+750x-1225= (x-5)[4x^2-90x+300+275/(x-5)]=0

Go from there....

basically take the first term and divide into equation's first term to start

x into 4x^3 = 4x^2 put over the dividend line as part of your answer(quotient)

multiply the 4x^2 by (x-5) in my case

Subtract your answer like in a long division problem

you should have canceled the first term and be left with -90x^2. Like a long division problem, bring down the next term +750x.

Now you have -90x^2+750x to divide by the (x-5)

Again divide the x into the -90x^2 to get -90x.

etc, etc...