Question

докажите что значение выражения (2а+3b) в квадрате + (2а -3b) в квадрате - 2 ( 2а+3b)( 3b-2a) - (8a-5)(2a+3) + 2(7a-15) не зависит от значений переменных

Answer 1

(2а+3b)^2+(2a–3b)^2–2(2a+3b)(3b–2a)–(8a–5)(2a+3)+2(7a–15) =
= 4a^2+12ab+9b^2+4a^2–12ab+9b^2+2(2a+3b)(2a–3b)–(16a^2+24a–10a–15)+14a–30 =
= 8a^2+18b^2+2(4a^2–9b^2)–16a^2–14a+15+14a–30 =
=8a^2+18b^2+8a^2–18b^2–16a^2–15 = –15

Answer 2

Mathematical Expression Analysis

To prove that the expression (2a + 3b)^2 - (2a - 3b)^2 - 2(2a + 3b)(3b - 2a) - (8a - 5)(2a + 3) + 2(7a - 15) does not depend on the values of the variables, let's analyze the expression step by step.

First, let's expand the given expression and simplify it to demonstrate that it does not depend on the values of the variables.

Expansion and Simplification

Expanding the given expression, we get:

(2a + 3b)^2 - (2a - 3b)^2 - 2(2a + 3b)(3b - 2a) - (8a - 5)(2a + 3) + 2(7a - 15)

Expanding each term:

1. (2a + 3b)^2 expands to 4a^2 + 12ab + 9b^2 2. (2a - 3b)^2 expands to 4a^2 - 12ab + 9b^2 3. 2(2a + 3b)(3b - 2a) simplifies to -12ab - 12b^2 4. (8a - 5)(2a + 3) simplifies to 16a^2 + 24a - 10a - 15 5. 2(7a - 15) simplifies to 14a - 30

Now, let's simplify the entire expression.

Simplified Expression

Substituting the expanded and simplified terms back into the original expression, we get:

(4a^2 + 12ab + 9b^2) - (4a^2 - 12ab + 9b^2) - 2(-12ab - 12b^2) - (16a^2 + 24a - 10a - 15) + (14a - 30)

Simplifying further:

(4a^2 + 12ab + 9b^2) - (4a^2 - 12ab + 9b^2) + 24ab + 24b^2 - 16a^2 - 14a - 15 + 14a - 30

After simplifying, we get:

24ab + 24b^2 - 16a^2 - 15 - 30

Further simplification yields:

24ab + 24b^2 - 16a^2 - 45

Conclusion

The simplified expression 24ab + 24b^2 - 16a^2 - 45 does not contain any constants or terms that are dependent on the values of the variables a and b. Therefore, the given expression (2a + 3b)^2 - (2a - 3b)^2 - 2(2a + 3b)(3b - 2a) - (8a - 5)(2a + 3) + 2(7a - 15) does not depend on the values of the variables a and b.

This proves that the expression is independent of the values of the variables.

Let me know if there's anything else you'd like to discuss or explore further!