Question

Докажите тождесвто. 1) (3x+4y)^2-(4y-3x)^2=48 xy 2) (1,5x-2y)^2+(2x+1,5y)^2 =6,25(x^2+y^2) Помогите пожалуйстаа)))

Answer 1

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Answer 2

Equation Simplification

To prove the given identity, let's simplify both sides of the equation step by step.

Starting with the left-hand side (LHS):

1. Expand the square of the binomial (3x + 4y)^2: ``` (3x + 4y)^2 = (3x)^2 + 2(3x)(4y) + (4y)^2 = 9x^2 + 24xy + 16y^2 ```

2. Expand the square of the binomial (4y - 3x)^2: ``` (4y - 3x)^2 = (4y)^2 - 2(4y)(3x) + (3x)^2 = 16y^2 - 24xy + 9x^2 ```

3. Subtract the result of step 2 from the result of step 1: ``` (3x + 4y)^2 - (4y - 3x)^2 = (9x^2 + 24xy + 16y^2) - (16y^2 - 24xy + 9x^2) = 9x^2 + 24xy + 16y^2 - 16y^2 + 24xy - 9x^2 = 48xy ```

Now let's simplify the right-hand side (RHS):

1. Expand the square of the binomial (1.5x - 2y)^2: ``` (1.5x - 2y)^2 = (1.5x)^2 - 2(1.5x)(2y) + (2y)^2 = 2.25x^2 - 6xy + 4y^2 ```

2. Expand the square of the binomial (2x + 1.5y)^2: ``` (2x + 1.5y)^2 = (2x)^2 + 2(2x)(1.5y) + (1.5y)^2 = 4x^2 + 6xy + 2.25y^2 ```

3. Add the result of step 1 and the result of step 2: ``` (1.5x - 2y)^2 + (2x + 1.5y)^2 = (2.25x^2 - 6xy + 4y^2) + (4x^2 + 6xy + 2.25y^2) = 6.25x^2 + 6xy + 6.25y^2 ```

Therefore, the simplified equation becomes:

48xy = 6.25x^2 + 6xy + 6.25y^2

Verifying the Identity

To verify the identity, we can compare the coefficients of the variables on both sides of the equation.

On the left-hand side (LHS), the coefficient of xy is 48.

On the right-hand side (RHS), the coefficient of xy is 6.

Since the coefficients are not equal, the given equation (1) + (3x + 4y)^2 - (4y - 3x)^2 = 48xy does not hold true.

Therefore, the given equation is not an identity.

Please let me know if there's anything else I can help you with!