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 =

The values of the variables are [tex]a = 6[/tex], [tex]b = 2[/tex], [tex]c = 6[/tex], [tex]d = 2[/tex], [tex]e = 6[/tex]. [tex]f = 6[/tex] and [tex]g = 1[/tex]

What are rational expressions?

Rational expressions are quotients of two algebraic expressions

The expression is given as:

[tex]\frac{2}{x^2 - 36} - \frac{1}{x^2 +6x} = \frac{2}{(x+6)(x-6)} - \frac{1}{x(x + a)}[/tex]

Factor out x in the expression

[tex]\frac{2}{x^2 - 36} - \frac{1}{x(x +6)} = \frac{2}{(x+6)(x-6)} - \frac{1}{x(x + a)}[/tex]

By comparison, we have:

[tex]x(x + 6) = x(x + a)[/tex]

This gives

[tex]a = 6[/tex]

Also, we have:

[tex]\frac{2}{x^2 - 36} - \frac{1}{x(x +6)} = \frac{bx}{(x+6)(x-6)x} - \frac{x-c}{(x+6)(x-6)x}[/tex]

Express x^2 - 36 as a difference of two squares

[tex]\frac{2}{(x + 6)(x - 6)} - \frac{1}{x(x +6)} = \frac{bx}{(x+6)(x-6)x} - \frac{x-c}{(x+6)(x-6)x}[/tex]

Rewrite as:

[tex]\frac{2x}{(x + 6)(x - 6)x} - \frac{x-6}{(x + 6)(x - 6)x} = \frac{bx}{(x+6)(x-6)x} - \frac{x-c}{(x+6)(x-6)x}[/tex]

By comparison

[tex]bx = 2x[/tex] and [tex]x - c = x - 6[/tex]

So, we have:

[tex]b = 2[/tex]

[tex]c = 6[/tex]

Also, we have:

[tex]\frac{2x}{(x + 6)(x - 6)x} - \frac{x-6}{(x + 6)(x - 6)x} = \frac{dx - x + e}{(x+6)(x-6)x}[/tex]

Take the LCM

[tex]\frac{2x - x + 6}{(x + 6)(x - 6)x} = \frac{dx - x + e}{(x+6)(x-6)x}[/tex]

By comparison, we have:

[tex]d = 2[/tex]

[tex]e = 6[/tex]

Solving further, we have:

[tex]\frac{2x - x + 6}{(x + 6)(x - 6)x} = \frac{x + f}{(x+6)(x-6)x}[/tex]

This gives

[tex]\frac{x + 6}{(x + 6)(x - 6)x} = \frac{x + f}{(x+6)(x-6)x}[/tex]

By comparison, we have:

[tex]f = 6[/tex]

Lastly, we have:

[tex]\frac{x + 6}{(x + 6)(x - 6)x} = \frac{g}{(x-6)x}[/tex]

This gives

[tex]\frac{1}{(x - 6)x} = \frac{g}{(x-6)x}[/tex]

By comparison,

[tex]g = 1[/tex]

Hence, the values of the variables are [tex]a = 6[/tex], [tex]b = 2[/tex], [tex]c = 6[/tex], [tex]d = 2[/tex], [tex]e = 6[/tex]. [tex]f = 6[/tex] and [tex]g = 1[/tex]