Question

50 POINTS!!!
I NEED STEPS

Answer 1

`(3)/(a-6)\quad \textsf{if}\;\;a \\eq -6,a \\eq 6`

Explanation:

Given expression:

`(a)/(a-6)-(3)/(a+6)+(a^2)/(36-a^2)`

Rewrite the third fraction:

`\implies (a^2)/(36-a^2)=(-a^2)/(-(36-a^2))=(-a^2)/(a^2-36)`

`\boxed{\begin{minipage}{4 cm}\underline{Difference of two squares }\\\\$x^2-y^2=(x-y)(x+y)$\\\end{minipage}}`

Apply the difference of two squares to the denominator of the third fraction:

`\implies a^2-36=a^2-6^2=(a-6)(a+6)`

Therefore the expression can be written as:

`\implies (a)/(a-6)-(3)/(a+6)+(-a^2)/((a-6)(a+6))`

`\implies (a)/(a-6)-(3)/(a+6)-(a^2)/((a-6)(a+6))`

The least common multiplier (LCM) of the denominator is (a - 6)(a + 6).

Adjust the fractions based on the LCM:

`\implies (a(a+6))/((a-6)(a+6))-(3(a-6))/((a-6)(a+6))-(a^2)/((a-6)(a+6))`

Simplify:

`\implies (a^2+6a)/((a-6)(a+6))-(3a-18)/((a-6)(a+6))-(a^2)/((a-6)(a+6))`

`\textsf{Apply the fraction rule} \quad (a)/(c)-(b)/(c)-(d)/(c)=(a-b-d)/(c):`

`\implies (a^2+6a-(3a-18)-a^2)/((a-6)(a+6))`

Simplify:

`\implies (3a+18)/((a-6)(a+6))`

Factor out 3 from the numerator:

`\implies (3(a+6))/((a-6)(a+6))`

Cancel the common factor (a + 6):

`\implies (3)/(a-6)`

Therefore:

`(a)/(a-6)-(3)/(a+6)+(a^2)/(36-a^2)=(3)/(a-6)\quad \textsf{if}\;\;a \\eq -6,a \\eq 6`