Solution - Nonlinear equations
Step by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "r2" was replaced by "r^2".
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(r2) - (22•3r23) = 0Step 2 :
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
r2 - 12r23 = -r2 • (12r21 - 1)
Trying to factor as a Difference of Cubes:
3.2 Factoring: 12r21 - 1
Theory : A difference of two perfect cubes, a3 - b3 can be factored into
(a-b) • (a2 +ab +b2)
Proof : (a-b)•(a2+ab+b2) =
a3+a2b+ab2-ba2-b2a-b3 =
a3+(a2b-ba2)+(ab2-b2a)-b3 =
a3+0+0-b3 =
a3-b3
Check : 12 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Equation at the end of step 3 :
-r2 • (12r21 - 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 : -r2 = 0
Multiply both sides of the equation by (-1) : r2 = 0
When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
r = ± √ 0
Any root of zero is zero. This equation has one solution which is r = 0
Solving a Single Variable Equation :
4.3 Solve : 12r21-1 = 0
Add 1 to both sides of the equation :
12r21 = 1
Divide both sides of the equation by 12:
r21 = 1/12 = 0.083
r = 21st root of (1/12)
The equation has one real solution
This solution is r = 21st root of ( 0.083) = 0.88840
Two solutions were found :
- r = 21st root of ( 0.083) = 0.88840
- r = 0
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