Step-by-step explanation:
Step by Step Solution:
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STEP
1
:
5
Simplify ———————
x2 - 81
Trying to factor as a Difference of Squares:
1.1 Factoring: x2 - 81
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 : 81 is the square of 9
Check : x2 is the square of x1
Factorization is : (x + 9) • (x - 9)
Equation at the end of step
1
:
(((x2)-18x)+81) 5
———————————————•———————————
(((x2)-6x)-27) (x+9)•(x-9)
STEP
2
:
x2 - 18x + 81
Simplify —————————————
x2 - 6x - 27
Trying to factor by splitting the middle term
2.1 Factoring x2 - 18x + 81
The first term is, x2 its coefficient is 1 .
The middle term is, -18x its coefficient is -18 .
The last term, "the constant", is +81
Step-1 : Multiply the coefficient of the first term by the constant 1 • 81 = 81
Step-2 : Find two factors of 81 whose sum equals the coefficient of the middle term, which is -18 .
-81 + -1 = -82
-27 + -3 = -30
-9 + -9 = -18 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -9 and -9
x2 - 9x - 9x - 81
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-9)
Add up the last 2 terms, pulling out common factors :
9 • (x-9)
Step-5 : Add up the four terms of step 4 :
(x-9) • (x-9)
Which is the desired factorization
Trying to factor by splitting the middle term
2.2 Factoring x2-6x-27
The first term is, x2 its coefficient is 1 .
The middle term is, -6x its coefficient is -6 .
The last term, "the constant", is -27
Step-1 : Multiply the coefficient of the first term by the constant 1 • -27 = -27
Step-2 : Find two factors of -27 whose sum equals the coefficient of the middle term, which is -6 .
-27 + 1 = -26
-9 + 3 = -6 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -9 and 3
x2 - 9x + 3x - 27
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-9)
Add up the last 2 terms, pulling out common factors :
3 • (x-9)
Step-5 : Add up the four terms of step 4 :
(x+3) • (x-9)
Which is the desired factorization
Canceling Out :
2.3 Cancel out (x-9) which appears on both sides of the fraction line.
Equation at the end of step
2
:
(x - 9) 5
——————— • —————————————————
x + 3 (x + 9) • (x - 9)
STEP
3
:
Canceling Out
3.1 Cancel out (x-9) which appears on both sides of the fraction line.
Final result :
5
—————————————————
(x + 3) • (x + 9)
