Roots of Cubic Equation: Finding $x_1,\,x_2$, and $x_3$

In summary, finding the roots of a cubic equation is important as it provides valuable information for solving real-world problems and understanding polynomial functions. The roots can be found using methods such as the rational root theorem, cubic formula, and factoring. The discriminant is significant in determining the number and nature of roots in a cubic equation. Not all cubic equations can be solved using algebraic methods, as some may have irrational or complex roots. However, cubic equations have practical applications in various fields, including engineering, physics, and economics.
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The roots $x_1,\,x_2$ and $x_3$ of the equation $x^3+ax+a=0$ where $a$ is a non-zero real number, satisfy $\dfrac{x_1^2}{x_2}+\dfrac{x_2^2}{x_3}+\dfrac{x_3^2}{x_1}=-8$. Find $x_1,\,x_2$ and $x_3$.
 
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We are given the following:
$x_1^3x_3+x_2^3x_1+x_3^3x_2=-8x_1x_2x_3,\\x_1+x_2+x_3=0,\\x_1x_2+x_2x_3+x_3x_1=a,\\x_1x_2x_3=-a$ and for $i=1,\,2,\,3$, $x_i^3+ax_i+a=0$.

Now

$x_1^3+ax_1+a=0\\x_2^3+ax_2+a=0\\x_3^3+ax_3+a=0$ gives

$(x_1^3x_3+x_2^3x_1+x_3^3x_2)+a(x_1x_3+x_2x_1+x_3x_2)+a(x_3+x_2+x_1)=0$

i.e. $8a+a^2=0,\implies a=-8$.

So the given equation is $x^3-8x-8=0$. One root is $-2$ and the other roots are given by $x^2-2x-4=0$, i.e. $x=1\pm \sqrt{5}$.
 

Related to Roots of Cubic Equation: Finding $x_1,\,x_2$, and $x_3$

1. What is a cubic equation?

A cubic equation is a polynomial equation of degree three, meaning it contains a variable raised to the third power. It can be written in the form ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants and x is the variable.

2. Why do we need to find the roots of a cubic equation?

Finding the roots of a cubic equation allows us to solve for the values of the variable x that make the equation true. This is useful in many mathematical and scientific applications, such as finding the intersection points of two curves or determining the maximum or minimum values of a function.

3. How do you find the roots of a cubic equation?

To find the roots of a cubic equation, we can use the cubic formula or the method of synthetic division. The cubic formula involves a long and complex equation, so the method of synthetic division is often preferred. This method involves dividing the cubic equation by a linear factor and then solving the resulting quadratic equation to find the roots.

4. Can a cubic equation have more than three roots?

No, a cubic equation can only have three roots. This is because a cubic equation can be factored into three linear factors, each representing a root. However, some of these roots may be complex numbers rather than real numbers.

5. How do you know if a cubic equation has real or complex roots?

A cubic equation will have real roots if the discriminant (b^2 - 4ac) is greater than or equal to 0. If the discriminant is less than 0, the equation will have complex roots. Additionally, if the coefficients of the cubic equation are all real numbers, the complex roots will be in conjugate pairs.

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