STEP
1
:
Equation at the end of step 1
(n • (c10)) - (2•11nc12) = 0
STEP
2
:
STEP
3
:
Pulling out like terms
3.1 Pull out like factors :
nc10 - 22nc12 = -nc10 • (22c2 - 1)
Trying to factor as a Difference of Squares:
3.2 Factoring: 22c2 - 1
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 22 is not a square !!
Ruling : Binomial can not be factored as the
difference of two perfect squares
Equation at the end of step
3
:
-nc10 • (22c2 - 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 -nc10 = 0
Setting any of the variables to zero solves the equation:
n = 0
c = 0
Solving a Single Variable Equation:
4.3 Solve : 22c2-1 = 0
Add 1 to both sides of the equation :
22c2 = 1
Divide both sides of the equation by 22:
c2 = 1/22 = 0.045
When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
c = ± √ 1/22
The equation has two real solutions
These solutions are c = ±√ 0.045 = ± 0.21320
Four solutions were found :
c = ±√ 0.045 = ± 0.21320
c = 0
n = 0