Question

(a + 3)/(a - 3) + (a - 3)/(a + 3)) * (3a ^ 2 + 27)/(9 - a ^ 2)

Answer 1

Ответ:

Приведем верхнюю дробь к общему знаменателю (а + 3) * (а – 3):

((а + 3) / (а – 3) + (а - 3) / (а + 3)) / ((3а2 + 27) / (9 – а2)) = ((а + 3) * (а + 3) / (а – 3) * (а + 3) + (а - 3) * (а – 3) / (а + 3) * (а – 3)) / ((3а2 + 27) / (9 – а2)) = ((а + 3) * (а + 3) + (а - 3) * (а – 3)) / ((а + 3) * (а – 3)) / ((3а2 + 27) / (9 – а2)) = ((а + 3)2 + (а - 3)2) / ((а + 3) * (а – 3)) / (3 * (а2 + 9)) / (9 – а2)).

Раскроем скобки в числителе верхней дроби и используем формулу разности квадратов для ее знаменателя:

(2а2 + 18) / (а2 – 9) / (3 * (а2 + 9)) / (9 – а2)) = - 2 * (а2 + 9) / (9 - а2) * ((9 – а2) / (3 * (а2 + 9))) = - 2/3.

ОТВЕТ: -2/3.

Answer 2

Ответ:

Решение и ответ на фото

Answer 3

To simplify the expression $\left(\frac{a + 3}{a - 3} + \frac{a - 3}{a + 3}\right) \cdot \frac{3a^2 + 27}{9 - a^2},$ you can follow these steps:

Step 1: Simplify each fraction individually:

a) Simplify $\frac{a + 3}{a - 3} + \frac{a - 3}{a + 3}$:

To add these fractions, we need a common denominator, which is $(a - 3)(a + 3) = a^2 - 9$. So, rewrite the fractions with this common denominator:

$\frac{(a + 3)^2}{(a - 3)(a + 3)} + \frac{(a - 3)^2}{(a - 3)(a + 3)}$

Now, add the fractions:

$\frac{(a + 3)^2 + (a - 3)^2}{(a - 3)(a + 3)}$

b) Simplify $\frac{3a^2 + 27}{9 - a^2}$:

Factor out 3 from the numerator:

$\frac{3(a^2 + 9)}{9 - a^2}$

Step 2: Combine the two fractions:

Now that both fractions have the same denominator, you can combine them into a single fraction:

$\frac{(a + 3)^2 + (a - 3)^2}{(a - 3)(a + 3)} \cdot \frac{3(a^2 + 9)}{9 - a^2}$

Step 3: Factor the numerator:

Factor the numerator, which is the sum of squares:

$(a + 3)^2 + (a - 3)^2 = a^2 + 6a + 9 + a^2 - 6a + 9$

Combine like terms:

$2a^2 + 18$

Step 4: Substitute back into the expression:

Now, substitute the factored numerator back into the expression:

$\frac{2a^2 + 18}{(a - 3)(a + 3)} \cdot \frac{3(a^2 + 9)}{9 - a^2}$

Step 5: Simplify further:

Simplify the expression further by canceling common factors:

$\frac{2(a^2 + 9)}{(a - 3)(a + 3)} \cdot \frac{3(a^2 + 9)}{-(a^2 - 9)}$

Now, cancel common factors between the numerators and denominators:

$\frac{2 \cdot 3 \cdot (a^2 + 9)}{(a - 3)(a + 3) \cdot -(a^2 - 9)}$

Simplify the constants:

$\frac{6(a^2 + 9)}{-(a^2 - 9)(a^2 - 9)}$

Step 6: Factor the denominators:

Factor the denominators further:

$-((a + 3)(a - 3)(a + 3)(a - 3))$

Step 7: Cancel common factors:

Now, cancel common factors between the numerators and denominators:

$\frac{6}{(a + 3)(a - 3)}$

So, the simplified expression is $\frac{6}{(a + 3)(a - 3)}$.