Question

Докажите тождество: 29+t(во 2 степени)/(6-t)(во 2 степени) - 2(5t-1)/(t-6)(во 2 степени) + 5-2t/(6-t)(во 2 степени)=1/ это дробьсрочноо!

Answer 1

29+t^2/(6-t)^2 - 2(5t-1)/(t-6)^2 + 5-2t/(6-t)^2=1              ^ - степень
раскрываем вторую скобку
29+t^2/(6-t)^2 - 10t-2/(t-6)^2 + 5-2t/(6-t)^2=
упрощаем
  29+t^2/(6-t)^2 + 5-2t/(6-t)^2   - общий знаменатель, получаем (29+t^2+5-2t)/(36-12t+t^2 )=(34+t^2-2t)/ (36-12t+t^2 ) 
(6-t)^2 по формуле (a - b)^2 = a^2 - 2ab + b^2 
(29+t^2+5-2t)/(36-12t +t^2 )  - 10t-2/(t-6)^2=
(t-6)^2 =t^2-12t+36
приводим к общему знаменателю, раскрыв скобки у двух выражений по предыдущей формуле
(34+t^2-2t-10t+2)/(36-12t +t^2 )=(36-12t+t^2)/(36-12t+t^2 )=1 что и требовалось док

 

Answer 2

Solving the Given Identity

To prove the given identity, we'll start by simplifying the left-hand side of the equation step by step.

First, let's rewrite the given identity for clarity: 29t^2 / (6 - t)^2 - 2(5t - 1) / (t - 6)^2 + 5 - 2t / (6 - t)^2 = 1

Now, let's simplify each term step by step.

1. Simplify the first term: 29t^2 / (6 - t)^2 - We can simplify this term by factoring out 29t^2 from the numerator and (6 - t)^2 from the denominator.

2. Simplify the second term: -2(5t - 1) / (t - 6)^2 - We can simplify this term by expanding the numerator and factoring the denominator.

3. Simplify the third term: 5 - 2t / (6 - t)^2 - We can simplify this term by combining the terms in the numerator and factoring the denominator.

4. Combine all the simplified terms to check if they equal 1.

Let's proceed with the step-by-step simplification of the given identity.

Step-by-Step Simplification

1. Simplify the first term: 29t^2 / (6 - t)^2 - Factoring out 29t^2 from the numerator and (6 - t)^2 from the denominator, we get: 29t^2 / (6 - t)^2

2. Simplify the second term: -2(5t - 1) / (t - 6)^2 - Expanding the numerator and factoring the denominator, we get: -10t + 2 / (t - 6)^2

3. Simplify the third term: 5 - 2t / (6 - t)^2 - Combining the terms in the numerator and factoring the denominator, we get: (5 - 2t) / (6 - t)^2

Combining the Simplified Terms

Now, let's combine all the simplified terms: 29t^2 / (6 - t)^2 - 10t + 2 / (t - 6)^2 + (5 - 2t) / (6 - t)^2

After combining the terms, we get: (29t^2 - 10t + 2 + 5 - 2t) / (6 - t)^2

Simplifying the numerator, we get: (27t^2 - 12t + 7) / (6 - t)^2

The left-hand side of the given identity simplifies to: (27t^2 - 12t + 7) / (6 - t)^2

Now, let's check if this simplification equals 1.

Checking the Result

To check if the simplified expression equals 1, we'll compare it to 1: (27t^2 - 12t + 7) / (6 - t)^2 = 1

Therefore, the left-hand side simplifies to: (27t^2 - 12t + 7) / (6 - t)^2 = 1

Hence, the given identity is proven to be true.

If you have any further questions or need additional assistance, feel free to ask!