Question

Simplify (x + 2/ x^2 + 2x -3) / (x + 2/x^2 - x)

Answer 1

`(x)/(x + 3)`

EXPLANATION

We want to simplify:

`\frac{x +2 }{ {x}^(2) + 2x - 3} / \frac{x + 2}{ {x}^(2)- x}`

Multiply by the reciprocal of the second fraction:

`\frac{x +2 }{ {x}^(2) + 2x - 3} * \frac{{x}^(2)- x}{ x + 2}`

Factor;

`(x +2 )/( (x + 3)(x - 1)) * (x(x - 1))/( x + 2)`

We cancel out the common factors to get:

`(x)/(x + 3)`

Answer 2

The simplest form is x/(x + 3)

Explanation:

* To simplify the rational Expression lets revise the factorization

of the quadratic expression

* To factor a quadratic in the form x² ± bx ± c:

- First look at the c term

# If the c term is a positive number, and its factors are r and s they

will have the same sign and their sum is b.

# If the c term is a negative number, then either r or s will be negative

but not both and their difference is b.

- Second look at the b term.

# If the c term is positive and the b term is positive, then both r and

s are positive.

Ex: x² + 5x + 6 = (x + 3)(x + 2)

# If the c term is positive and the b term is negative, then both r and s

are negative.

Ex: x² - 5x + 6 = (x -3)(x - 2)

# If the c term is negative and the b term is positive, then the factor

that is positive will have the greater absolute value. That is, if

|r| > |s|, then r is positive and s is negative.

Ex: x² + 5x - 6 = (x + 6)(x - 1)

# If the c term is negative and the b term is negative, then the factor

that is negative will have the greater absolute value. That is, if

|r| > |s|, then r is negative and s is positive.

Ex: x² - 5x - 6 = (x - 6)(x + 1)

* Now lets solve the problem

- We have two fractions over each other

- Lets simplify the numerator

∵ The numerator is
`(x+2)/(x^(2)+2x-3)`

- Factorize its denominator

∵ The denominator = x² + 2x - 3

- The last term is negative then the two brackets have different signs

∵ 3 = 3 × 1

∵ 3 - 1 = 2

∵ The middle term is +ve

∴ -3 = 3 × -1 ⇒ the greatest is +ve

∴ x² + 2x - 3 = (x + 3)(x - 1)

∴ The numerator =
`((x+2))/((x+3)(x-2))`

- Lets simplify the denominator

∵ The denominator is
`(x+2)/(x^(2)-x)`

- Factorize its denominator

∵ The denominator = x² - 2x

- Take x as a common factor and divide each term by x

∵ x² ÷ x = x

∵ -x ÷ x = -1

∴ x² - 2x = x(x - 1)

∴ The denominator =
`((x+2))/(x(x-1))`

* Now lets write the fraction as a division

∴ The fraction =
`(x+2)/((x+3)(x-1))` ÷
`(x+2)/(x(x-1))`

- Change the sign of division and reverse the fraction after it

∴ The fraction =
`((x+2))/((x+3)(x-1))*(x(x-1))/((x+2))`

* Now we can cancel the bracket (x + 2) up with same bracket down

and cancel bracket (x - 1) up with same bracket down

∴ The simplest form =
`(x)/(x+3)`