Question

Enter the values for the highlighted variables that show how to subtract the rational expressions correctly:

StartFraction 2 Over x squared minus 36 EndFraction minus StartFraction 1 Over x squared + 6 x EndFraction = StartFraction 2 Over (x + 6) (x minus 6) EndFraction minus StartFraction 1 Over x (x + a) EndFraction. = StartFraction b x Over (x + 6) (x minus 6) x EndFraction minus StartFraction x minus c Over (x + 6) (x minus 6) x EndFraction. = StartFraction d x minus x + e Over (x + 6) (x minus 6) x EndFraction. = StartFraction x + f Over (x + 6) (x minus 6) x EndFraction. = StartFraction g Over x (x minus 6) EndFraction

a =

b =

c =

d =

e =

f =

g =

Answer 1

a=6, b=2, c=6, d=2, e=6, f=6, g=1

Explanation:

You want the values of letters a–g that show the correct simplification of ...

`(2)/(x^2-36)-(1)/(x^2+6x)=(2)/((x+6)(x-6))-(1)/(x(x+a))\\\\=(bx)/((x+6)(x-6)x)-(x-c)/((x+6)(x-6)x)=(dx-x+e)/((x+6)(x-6)x)\\\\=(x+f)/((x+6)(x-6)x)=(g)/(x(x+6))`

Solution

The difference of squares is factored as (a² -b²) = (a +b)(a -b), so the denominator of the first term can be factored. The common factor x can be factored out of the denominator of the second term. This gives ...

`(2)/(x^2-36)-(1)/(x^2+6x)= (2)/((x+6)(x-6))-(1)/(x(x+6))\quad a=6`

Multiplying the first term by x/x and the second term by (x-6)/(x-6) gives ...

`(2x)/((x+6)(x-6)x)-(x-6)/((x+6)(x-6)x)\quad b=2,\ c=6`

Writing the sum over the common denominator, we have ...

`(2x-x+6)/((x+6)(x-6)x)\quad d=2,\ e=6`

Collecting terms in the numerator gives ...

`(x+6)/((x+6)(x-6)x)\quad f=6`

Finally, cancelling the common factor (x+6) from numerator and denominator, we have the simplified form ...

`(1)/(x(x+6))\quad g=1`

The values of the letters are ...

a=6, b=2, c=6, d=2, e=6, f=6, g=1