Question

How do you solve $3{{x}^{3}}-2{{x}^{2}}-12x+8=0$ ?

Answer 1

Hint: In this question, we have to find the value of x. As we have to solve the cubic equation, it implies we will get three values of x. Therefore, we will solve this problem by using factorization. We will separate the equation into two polynomials and then take common out the variable or the constant from each polynomial. Thus, we get three different equations; therefore we will solve the three equations, to get the required answer for the problem.

Complete step by step answer:
According to the question, we have to find the value of x.
Thus, we will use the factorization method to get the solution to the problem.
The equation given to us is $3{{x}^{3}}-2{{x}^{2}}-12x+8=0$ ---------- (1)
We will first separate the equation (1) into two polynomials, we get
$(3{{x}^{3}}-2{{x}^{2}})+(-12x+8)=0$
Now, we will take ${{x}^{2}}$ common from the first polynomial and -4 from the next polynomial, we get
${{x}^{2}}(3x-2)-4(3x-2)=0$
So, we will take common (3x-2) from the above equation, we get
$(3x-2)({{x}^{2}}-4)=0$
Now, we will use the algebraic identity $(a-b)(a+b)={{a}^{2}}-{{b}^{2}}$ in the above equation, we get
$(3x-2)(x-2)(x+2)=0$
Therefore, we will get three different equations from the above equation, we get
$(3x-2)=0$ -------- (2)
$(x-2)=0$ ------- (3)
$(x+2)=0$ ------- (4)
Now, we will solve equation (2), which is
$(3x-2)=0$
Now, add 2 on both sides of the above equation, we get
$3x-2+2=0+2$
As we know, the same terms cancel out each other, therefore we get
$3x=2$
Now, we will divide 3 on both sides of the above equation, we get
$\dfrac{3}{3}x=\dfrac{2}{3}$
Therefore we get
$x=\dfrac{2}{3}$
Now, we will solve equation (3), which is
$x-2=0$
Now, add 2 on both sides of the above equation, we get
$3x-2+2=0+2$
As we know, the same terms cancel out each other, therefore we get
$x=2$
Now, we will solve equation (4), which is
$x+2=0$
Now, subtract 2 on both sides of the above equation, we get
$x+2-2=0-2$
As we know, the same terms cancel out each other, therefore we get
$x=-2$

Therefore, for the equation $3{{x}^{3}}-2{{x}^{2}}-12x+8=0$ , the value of x is $\dfrac{2}{3},2,-2$

Note: While solving this problem, keep in mind the steps you are using to avoid confusion and mathematical errors. One of the alternative methods to solve this problem is using the hidden-trial method, after that, we will use the long division method to get a quadratic equation, and in the end solve the quadratic equation using splitting the middle term method.