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Step by Step Solution

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "x7" was replaced by "x^7".

Rearrange:

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

2*x^2-2*x^7-(5)=0

Step 1 :

Equation at the end of step 1 :

  ((2 • (x²)) -  2x⁷) -  5  = 0 
 Step  2  :

Equation at the end of step 2 :

  (2x² -  2x⁷) -  5  = 0

Step 3 :

Step 4 :

Pulling out like terms :

4.1 Pull out like factors :

-2x⁷ + 2x² - 5 = -1 • (2x⁷ - 2x² + 5)

Polynomial Roots Calculator :

4.2    Find roots (zeroes) of :       F(x) = 2x⁷ - 2x² + 5
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 2 and the Trailing Constant is 5.

The factor(s) are:

of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,5

Let us test ....

P	Q	P/Q	F(P/Q)
-1	1	-1.00	1.00
-1	2	-0.50	4.48
-5	1	-5.00	-156295.00
-5	2	-2.50	-1228.20
1	1	1.00	5.00
1	2	0.50	4.52
5	1	5.00	156205.00
5	2	2.50	1213.20

Polynomial Roots Calculator found no rational roots

Equation at the end of step 4 :

  -2x⁷ + 2x² - 5  = 0

Step 5 :

Equations of order 5 or higher :

5.1 Solve -2x⁷+2x²-5 = 0

Points regarding equations of degree five or higher.

(1) There is no general method (Formula) for solving polynomial equations of degree five or higher.

(2) By the Fundamental theorem of Algebra, if we allow complex numbers, an equation of degree n will have exactly n solutions
(This is if we count double solutions as 2 , triple solutions as 3 and so on

) (3) By the Abel-Ruffini theorem, the solutions can not always be presented in the conventional way using only a finite amount of additions, subtractions, multiplications, divisions or root extractions

(4) If F(x) is a polynomial of odd degree with real coefficients, then the equation F(X)=0 has at least one real solution.

(5) Using methods such as the Bisection Method, real solutions can be approximated to any desired degree of accuracy. Failed to find the initial interval for implementing the BiSection Method