Simplify to the lowest terms (2x^2-22x)/(x^2-10x-11)

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sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You should write the factored form of denominator, hence, you need to find the zeroes of denominator such that:

`x^2-10x-11 = 0`

You need to use quadratic formula such that:

`x_(1,2) = (10+-sqrt(100 + 4*11))/2`

`x_(1,2) = (10+-sqrt144)/2`

`x_(1,2) = (10+-12)/2`

`x_1 = 11 ; x_2 = -1`

You may write the factored form of denominator such that:

`x^2-10x-11 = (x-11)(x+1)`

You need to factor out `2x`  to numerator such that:

`2x^2-22x = 2x(x-11)`

You need to substitute `2x(x-11)`  for `2x^2-22x`  and `(x-11)(x+1)`  for x^2-10x-11 such that:

`(2x^2-22x)/(x^2-10x-11) = (2x(x-11))/((x-11)(x+1))`

Reducing by factor `x-11`  yields:

`(2x^2-22x)/(x^2-10x-11) = 2x/(x+1)`

Hence, simplifying the fraction to its lowest terms yields `(2x^2-22x)/(x^2-10x-11) = 2x/(x+1).`

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giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll factor the numerator by 2x:

2x(x-11)/(x^2-10x-11)

We'll recall that quadratic is the result of the product of two linear factors.

We'll find the roots of quadratic:

x^2-10x-11 = 0

We know that a quadratic may be written when knowing the sum and the product of the roots:

x^2 - Sx + P = 0
Comparing, we'll get:
S = 10 and P = -11 => the roots are x1 = 11 and x2 = -1

x^2-10x-11 = (x-11)(x+1)

We'll re-write the fraction:

2x(x-11)/(x-11)(x+1)

We'll divide both numerator and denominator by (x-11) and we'll get:

2x/(x+1)

The given fraction simplified to the lowest terms is 2x/(x+1).

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