Question

Simplifying rational expressions
x^2-16/x^2+6x+8 = x-4/x+2
x^2-x-6/x^2-3x-10 = x-3/x-5

Answer 1

Answers are:

x-4/x+2

x-3/x-5

Explanation:

Answer 2

1)

`(x^2-16)/(x^2+6x+8)`

Decompose the numerator and denominator into multipliers

To simplify the numerator we use the formula of difference of squares

`x^2-y^2=(x-y)(x+y)`

`x^2-16=(x-4)(x+4)`

To decompose the denominator into multipliers solve the square equation

`x^2+6x+8=0\\D=6^2-4*8=4=2^2\\x_1=(-6+2)/(2) =-2\\x_2=(-6-2)/(2) =-4`

Formula for factoring a square equation

`(x-x_1)(x-x_2)`

Substituting the found roots of the equation into the formula

`(x-(-2))(x-(-4))=(x+2)(x+4)`

After simplifying the numerator and denominator we get a fraction

`((x-4)(x+4))/((x+2)(x+4))=(x-4)/(x+2)`, so

`(x^2-16)/(x^2+6x+8)=((x-4)(x+4))/((x+2)(x+4))=(x-4)/(x+2)`

2)

`(x^2-x-6)/(x^2-3x-10)`

Decompose the numerator and denominator into multipliers

To decompose the numerator into multipliers solve the square equation

`x^2-x-6=0\\D=(-1)^2-4*(-6)=25=5^2\\x_1=(1+5)/(2) =3\\x_2=(1-5)/(2) =-2`

Formula for factoring a square equation

`(x-x_1)(x-x_2)`

Substituting the found roots of the equation into the formula

`(x-3)(x-(-2))=(x-3)(x+2)`

To decompose the denominator into multipliers solve the square equation

`x^2-3x-10=0\\D=(-3)^2-4*(-10)=49=7^2\\x_1=(3+7)/(2) =5\\x_2=(3-7)/(2) =-2`

Formula for factoring a square equation

`(x-x_1)(x-x_2)`

Substituting the found roots of the equation into the formula

`(x-5)(x-(-(-2))=(x-5)(x+2)`

After simplifying the numerator and denominator we get a fraction

`((x-3)(x+2))/((x-5)(x+2))=(x-3)/(x-5)`, so

`(x^2-x-6)/(x^2-3x-10)=((x-3)(x+2))/((x-5)(x+2))=(x-3)/(x-5)`

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