Encounter a cubic equation: $4x^3-5x^2+x-14=0$

Question

$$y=2x-1\tag1$$

$$xy^2-x^2=14\tag2$$

Solving for $x$ and $y$ via the method of substitution

$$x(2x-1)^2-x^2=14\tag3$$

$$4x^3-5x^2+x-14=0\tag4$$

How do I continue $(4)?$ Is there an easy method solving cubic equation?

Answer 1

As @Barry Cipra points out, you can try some small integer values of $x$ and a root will pop right out. However, in my opinion, this is kinda cheating since we're not always guaranteed a nice simple small integer root for problems like this. However, there are a couple of tricks that can lead you to the root.

Let $p(x)=4x^3-5x^2+x-14$. Then $p'(x)=12x^2-10x+1$, which has roots at $x=\frac5{12}\pm\frac{\sqrt{13}}{12}$. Hence, $p'(x)$ is negative only at $[0.116,0.717]$. So $p(x)$ is ascending for all negative $x$, but $p(0)<0$, so there are no negative roots. Since we are guaranteed at least $1$ real root for every cubic equation, $p(x)$ has at least $1$ positive real root.

You could use numerical or graphical methods to approximate the root (and then test nearby values, to confirm the root).

We could try the substitution $x=z-a$, where $a$ is a constant. Then we can simplify the polynomial by taking out the $x^2$ term: $$\begin{aligned} p(x)&=4\left(z^3-3az^2+3a^2z-a^3\right)-5\left(z^2-2az+a^2\right)+(z-a)-14\\ &=4z^3+\left(-12a-5\right)z^2+(12a^2+10a+1)z+(-4a^3-5a^2-a-14)\\ &=4z^3-\frac{13}{12}z-\frac{3059}{216}\quad\quad a=-5/12\\ \end{aligned}$$

We could then proceed with an algorithm for solving cubic equations.