Solution - Nonlinear equations
Step by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "x9" was replaced by "x^9".
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(x2) - (2•3x9) = 0Step 2 :
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
x2 - 6x9 = -x2 • (6x7 - 1)
Polynomial Roots Calculator :
3.2 Find roots (zeroes) of : F(x) = 6x7 - 1
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 -1.
The factor(s) are:
of the Leading Coefficient : 1,2 ,3 ,6
of the Trailing Constant : 1
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -7.00 | ||||||
| -1 | 2 | -0.50 | -1.05 | ||||||
| -1 | 3 | -0.33 | -1.00 | ||||||
| -1 | 6 | -0.17 | -1.00 | ||||||
| 1 | 1 | 1.00 | 5.00 | ||||||
| 1 | 2 | 0.50 | -0.95 | ||||||
| 1 | 3 | 0.33 | -1.00 | ||||||
| 1 | 6 | 0.17 | -1.00 |
Polynomial Roots Calculator found no rational roots
Equation at the end of step 3 :
-x2 • (6x7 - 1) = 0
Step 4 :
Theory - Roots of a product :
4.1 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.
Solving a Single Variable Equation :
4.2 Solve : -x2 = 0
Multiply both sides of the equation by (-1) : x2 = 0
When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
x = ± √ 0
Any root of zero is zero. This equation has one solution which is x = 0
Solving a Single Variable Equation :
4.3 Solve : 6x7-1 = 0
Add 1 to both sides of the equation :
6x7 = 1
Divide both sides of the equation by 6:
x7 = 1/6 = 0.167
x = 7th root of (1/6)
The equation has one real solution
This solution is x = 7th root of ( 0.167) = 0.77417
Two solutions were found :
- x = 7th root of ( 0.167) = 0.77417
- x = 0
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