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Step by Step Solution

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "x4" was replaced by "x^4".

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

x^2*x-12-(x^4)=0

Step 1 :

Step 2 :

Pulling out like terms :

2.1 Pull out like factors :

-x⁴ + x³ - 12 = -1 • (x⁴ - x³ + 12)

Polynomial Roots Calculator :

2.2    Find roots (zeroes) of :       F(x) = x⁴ - x³ + 12
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case, the Leading Coefficient is 1 and the Trailing Constant is 12.

The factor(s) are:

of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,12

Let us test ....

P	Q	P/Q	F(P/Q)
-1	1	-1.00	14.00
-2	1	-2.00	36.00
-3	1	-3.00	120.00
-4	1	-4.00	332.00
-6	1	-6.00	1524.00
-12	1	-12.00	22476.00
1	1	1.00	12.00
2	1	2.00	20.00
3	1	3.00	66.00
4	1	4.00	204.00
6	1	6.00	1092.00
12	1	12.00	19020.00

Polynomial Roots Calculator found no rational roots

Equation at the end of step 2 :

  -x⁴ + x³ - 12  = 0

Step 3 :

Quartic Equations :

3.1 Solve -x⁴+x³-12 = 0

In search of an interavl at which the above polynomial changes sign, from negative to positive or the other wayaround.

Method of search: Calculate polynomial values for all integer points between x=-20 and x=+20

No interval at which a change of sign occures has been found. Consequently, Bisection Approximation can not be used. As this is a polynomial of an even degree it may not even have any real (as opposed to imaginary) roots