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Solution - Nonlinear equations

t = 0.0000 - 1.5811 i

t=0.0000-1.5811i

t = 0.0000 + 1.5811 i

t=0.0000+1.5811i

Step by Step Solution

Rearrange:

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

0-(16*t^2+40)=0

Step by step solution :

Step 1 :

Equation at the end of step 1 :

  0 -  (2⁴t² +  40)  = 0

Step 2 :

Step 3 :

Pulling out like terms :

3.1 Pull out like factors :

-16t² - 40 = -8 • (2t² + 5)

Polynomial Roots Calculator :

3.2    Find roots (zeroes) of :       F(t) = 2t² + 5
Polynomial Roots Calculator is a set of methods aimed at finding values of  t  for which   F(t)=0

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers t 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	7.00
-1	2	-0.50	5.50
-5	1	-5.00	55.00
-5	2	-2.50	17.50
1	1	1.00	7.00
1	2	0.50	5.50
5	1	5.00	55.00
5	2	2.50	17.50

Polynomial Roots Calculator found no rational roots

Equation at the end of step 3 :

  -8 • (2t² + 5)  = 0

Step 4 :

Equations which are never true :

4.1 Solve : -8 = 0

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

Solving a Single Variable Equation :

4.2      Solve  :    2t²+5 = 0

Subtract 5 from both sides of the equation :
                      2t² = -5
Divide both sides of the equation by 2:
                     t² = -5/2 = -2.500

When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
                      t = ± √ -5/2

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

Accordingly, √ -5/2 =
                    √ -1• 5/2   =
                    √ -1 •√ 5/2 =
                    i •  √ 5/2

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

                      t= 0.0000 + 1.5811 i
                      t= 0.0000 - 1.5811 i

Two solutions were found :

t= 0.0000 - 1.5811 i
t= 0.0000 + 1.5811 i

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