# The Cubic Formula: ax^3+bx^2+cx+d=0

• Bill Foster
In summary, the conversation discusses the process of using Cardano's formula to find roots of a cubic equation. To do this, the equation must be reduced to the form x^3+mx=n and then solved using the quadratic formula. An example is given and the steps are explained in detail to show how to find the roots of the equation.
Bill Foster
The quadratic formula is, as you are well aware:

$$ax^2+bx+c=0$$ $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

What is the cubic formula?

$$ax^3+bx^2+cx+d=0$$

Thanks.

Cardano's formula will get you one root, but you have to transform the general equation a*x^3+b*x^2+cx+d=0 into e*z^3+f*z+g=0 by use of the substitution z=x-b/3. Mathworld explains a bit more: http://mathworld.wolfram.com/CubicFormula.html

Thanks.

could someone just show me with an example on how to work the cardano and Del ferro method from start to finish say using x^3-6x^2+2x-1

The first thing you do is "reduce" the equation to the form $x^3+ bx= c$ without any $x^2$ term. To do that, let x= y- a. Then $x^3= (y- a)^3= y^3- 3ay^2+ 3a^2y- a^3[/math] and [itex]x^2= (y- a)^2= y^2- 2ay+ a^2[/math] $$x^3- 6x^2+ 2x-1= y^3 -ay^2+ a^2y- a^3- 6y^2+ 12ay- 6a^2+ 2y- 2a- 1$$$$= y^3+ (-a- 6)y^2+ (a^2+ 2)y+ (-a^3- 2a- 1)[/math] That will have no "[itex]y^2$" term is a= -6 and, in that case, the polynomial is $y^3+ 38y+ 229$ and so our equation is $y^3+ 38y+ 229= 0$. Here's a quick review of Cardano's formula: $(a+ b)^3= a^3+ 3a^2b+ 3ab^2+ b^3$ $-3ab(a+ b)= -3a^2b- 3ab^3$ so that $(a+ b)^3+ 3ab(a+ b)= a^3+ b^3$. In particular, if we let x= a+b, m= 3ab, and $n= a^3+ b^3$, $x^3+ mx= n$. Can we go the other way? That is, given m and n, can we find a and b and so find x? Yes, we can. From $m= 3ab$, we have $b= m/(3a)$ so $a^3+ b^3= a^3+ m^3/(3^2a^3)= n$. Multiplying through by $a^3$, $(a^3)^2+ m^3/3^3= na^3$ or $(a^3)^2- na^3+ m^3/3^3= 0$. We can think of that as a quadratic equation in $a^3$ and solve it with the quadratic formula: [tex]a^3= \frac{n\pm\sqrt{n^2- 4\frac{m^3}{3^3}}}{2}[/itex]$= \frac{n}{2}\pm\sqrt{\left(\frac{n}{2}\right)^2- \left(\frac{m}{3}\right)^3}$$ and a is the cube root of that. From [itex]n= a^3+ b^3$ we have
$$b^3= n- a^3= \frac{n}{2}\mp\sqrt{\left(\frac{n}{2}\right)^2- \left(\frac{m}{3}\right)^3}$$.

Now, in this problem, $y^3+ 38y+ 229= 0$ or $y^3+ 38y= -229$ so m= 38 and n= -229.
$$\frac{n}{2}= -\frac{229}{2}$$
and
$$\left(\frac{n}{2}\right)^2= \frac{52441}{4}[/itex] [tex]\frac{m}{3}= \frac{38}{3}$$
and
$$\left(\frac{m}{3}\right)^3= \frac{54872}{27}$$

So
$$a^3= -\frac{229}{2}\pm\sqrt{\frac{52441}{4}- \frac{54872}{27}}$$
and
$$b^3= -\frac{229}{2}\mp\sqrt{\frac{52441}{4}- \frac{54872}{27}}$$

Calculate those numbers, take the cube roots to find a and b and then find x= a+ b.
Each of those will have 3 cube roots but in the various ways of combining them, some things will cancel so that there will be, at most, 3 roots to the equation.

thank yo for taking the time to show me this but I am still confused, after plugging in y-a into the equation how does a= -6

## 1. What is the Cubic Formula?

The Cubic Formula is a mathematical equation used to find the roots or solutions of a cubic polynomial equation in the form of ax^3+bx^2+cx+d=0. It provides a way to solve cubic equations that cannot be easily factored or solved by other methods.

## 2. How is the Cubic Formula derived?

The Cubic Formula is derived using a technique called the Cardano method, which was developed by Italian mathematician Gerolamo Cardano in the 16th century. It uses a combination of substitution, factoring, and simplification to obtain the final formula.

## 3. What are the three possible solutions of the Cubic Formula?

The Cubic Formula can have three possible solutions, which are called roots or solutions. These solutions can be real numbers or complex numbers. If the discriminant of the equation is positive, the three solutions will be real numbers. If the discriminant is negative, the three solutions will be complex numbers.

## 4. How do you use the Cubic Formula to solve an equation?

To use the Cubic Formula to solve an equation, you need to first rearrange the equation into the form of ax^3+bx^2+cx+d=0. Then, plug in the values of a, b, c, and d into the formula and simplify to obtain the three solutions. These solutions can then be substituted back into the original equation to check for accuracy.

## 5. What are some real-life applications of the Cubic Formula?

The Cubic Formula has various real-life applications, such as in physics, engineering, and finance. It can be used to solve equations related to projectile motion, fluid dynamics, and population growth. In engineering, the Cubic Formula can be used to find the roots of equations related to circuit analysis and structural design. In finance, it can be used to calculate interest rates and investment returns.

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