{x^2+6x-27/x^2-9x+18
how to solve step by step

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)

f is the midpoint of