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Solution - Nonlinear equations

x = 0.0000 - 1.3540 i

x=0.0000-1.3540i

x = 0.0000 + 1.3540 i

x=0.0000+1.3540i

Step by Step Solution

Step by step solution :

Step 1 :

Equation at the end of step 1 :

  22 +  (2²•3x²)  = 0

Step 2 :

Step 3 :

Pulling out like terms :

3.1 Pull out like factors :

12x² + 22 = 2 • (6x² + 11)

Polynomial Roots Calculator :

3.2    Find roots (zeroes) of :       F(x) = 6x² + 11
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 6 and the Trailing Constant is 11.

The factor(s) are:

of the Leading Coefficient : 1,2 ,3 ,6
of the Trailing Constant : 1 ,11

Let us test ....

P	Q	P/Q	F(P/Q)
-1	1	-1.00	17.00
-1	2	-0.50	12.50
-1	3	-0.33	11.67
-1	6	-0.17	11.17
-11	1	-11.00	737.00
-11	2	-5.50	192.50
-11	3	-3.67	91.67
-11	6	-1.83	31.17
1	1	1.00	17.00
1	2	0.50	12.50
1	3	0.33	11.67
1	6	0.17	11.17
11	1	11.00	737.00
11	2	5.50	192.50
11	3	3.67	91.67
11	6	1.83	31.17

Polynomial Roots Calculator found no rational roots

Equation at the end of step 3 :

  2 • (6x² + 11)  = 0

Step 4 :

Equations which are never true :

4.1 Solve : 2 = 0

This equation has no solution.
A a non-zero constant never equals zero.

Solving a Single Variable Equation :

4.2      Solve  :    6x²+11 = 0

Subtract 11 from both sides of the equation :
                      6x² = -11
Divide both sides of the equation by 6:
                     x² = -11/6 = -1.833

When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
                      x = ± √ -11/6

In Math,  i  is called the imaginary unit. It satisfies   i²  =-1. Both   i   and   -i   are the square roots of   -1

Accordingly, √ -11/6 =
                    √ -1• 11/6   =
                    √ -1 •√ 11/6 =
                    i •  √ 11/6

The equation has no real solutions. It has 2 imaginary, or complex solutions.

                      x= 0.0000 + 1.3540 i
                      x= 0.0000 - 1.3540 i

Two solutions were found :

x= 0.0000 - 1.3540 i
x= 0.0000 + 1.3540 i

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