Difference of two squares.

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Page Overview:

• Column Multiplication Method yields (C+A)(C-A) =  C² - A²  
• Zero Product Rule:  If a product equals zero then at least one of its factors equals zero - that is equivalent too: If all the factors in a product are non-zero, then the product is non-zero

• Two ways to solve x² - A²  = 0 - using square route or by factoring & applying the zero product rule.

The following column multiplication shows why the difference of two squares identity hold

(C+A)(C-A) =  C² - A²   

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Column Multiplication

C + A
C  - A  (times)
CA + C²
-AC  - A²   (add)
    C² - A² 

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The identity

(C+A)(C-A) =  C² - A²   

in equivalent form

C² - A² = (C+A)(C-A)

shows how the difference of two squares may be factored.

In practice, the application of this identity to expressions requires identification of the C² and - A² parts of the expression. 

Example (I) - Applied to Numbers

36 - 16

= 6² - 4²            C² - A² 
= (6+4)(6-4)      (C+A)(C-A)
= (10)(2)
= 20

Example (II) - Applied to Expressions

x² - 25 

= x² - 5²            C² - A² 
= (x+5)(x-5)      (C+A)(C-A)

Example (III) - Applied to Expressions

(2x-3)² - 100 

=(2x-3)² - 10²            C² - A² 
= ((2x-3)+10)((2x-3) -10)     (C+A)(C-A)
= (2x+7)(2x-13)

The Zero Product Law

Exercise - or food for thought.:

Find the value of the missing digits ABC in the following puzzle

234
ABC (times)
000
000
000 (add)
000

Assumption: If all the factors in a  product are nonzero then the product is nonzero.

The equivalent contrapositive form of this assumed implication rule is the following: Zero Product Law

If a product equals zero then at least one of its factors equals zero

The zero product law, its direct or indirect assumption,  provides a reason for factoring quadratics and further polynomials p(x) for which the question where does p(x) = 0 is of interest.

Two ways to solve equations:

Example A.

Suppose we want to solve

0 = x² - 25 = (x+5)(x-5).

The latter was shown above in example II

The solve an equation route would be to observe

 x² = 25 requires x = 5 or x = -5
since 5 = the principal square root of 25.

The factorization route may be to observe that both factors in the product (x+5)(x-5)  = x² - 25 will be nonzero when x does not have the values + 5 or -5. The only way that one of the factors and hence the product can be zero is if x = 5 or x = -5.

Example B.

Suppose we want to solve

0 = (2x-3)² - 100 =  (2x+7)(2x-13)

The latter was shown above in example III.

The equation route to solve is say

(2x-3)² =  100 requires

Check that for x = 6.5 and x = -3.5, both satisfy

0 = (2x-3)² - 100  or (2x-3)² =  100

Now the factoring route observes

(2x-3)² - 100 =  (2x+7)(2x-13)

and that the product can only be zero if and only if

 2x+ 7 = 0 or 2x - 13 = 0

or equivalently, if and only if,

2x = -7 or 2x = 13

or equivalently, if and only if,

x = -7/2 = -3.5 or  x = 13/2 = 6.5

This is the same result as before, except for the order of the solutions, an immaterial matter.

In solving quadratic and other polynomial equations, you have a choice between the factorization route and the equation solving route. Choose the route that is most convenient if that is permitted by your instructor.

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