Question

Решить уравнение,применив формулу сокращённого уравнения 1) (2х+1)^2-4х^2=9. 2) (2х-3)^2=9+4(Х-3)(Х+3). 3)(Х-5)^2=5х^2-(2х-1)(2х+1). 4) (3х+2)(3х-2)-32=9(Х-2)^2. 5)(2х-1)(2х+1)-4(х+5)^2=19. 6)(6а-1)(6а+1)= 4а(9а+2)-1. 7) (3х-1)^2-9х^2=10. 8) (2х+1)^2-3(Х-5)^2=(Х+3)(Х-3)

Answer 1

1. (2x+1)^2 - 4x^2 = 9

Expanding the square: 4x^2 + 4x + 1 - 4x^2 = 9

Simplifying: 4x + 1 = 9

Subtracting 1 from both sides: 4x = 8

Dividing by 4: x = 2

So the solution is x = 2.

2. (2x-3)^2 = 9 + 4(x-3)(x+3)

Expanding the square: 4x^2 - 12x + 9 = 9 + 4(x^2 - 9)

Simplifying: 4x^2 - 12x + 9 = 9 + 4x^2 - 36

Combining like terms: -12x = -36

Dividing by -12: x = 3

So the solution is x = 3.

3. (x-5)^2 = 5x^2 - (2x-1)(2x+1)

Expanding the square and simplifying: x^2 - 10x + 25 = 5x^2 - (4x^2 - 1)

Distributing the negative sign: x^2 - 10x + 25 = 5x^2 - 4x^2 + 1

Combining like terms: x^2 - 10x + 25 = x^2 + 1

Subtracting x^2 from both sides: -10x + 25 = 1

Subtracting 25 from both sides: -10x = -24

Dividing by -10: x = 2.4

So the solution is x = 2.4.

4. (3x+2)(3x-2) - 32 = 9(x-2)^2

Expanding and simplifying: 9x^2 - 4 - 32 = 9(x^2 - 4x + 4)

Combining like terms: 9x^2 - 36 = 9x^2 - 36x + 36

Subtracting 9x^2 from both sides: -36 = -36x + 36

Subtracting 36 from both sides: -72 = -36x

Dividing by -36: x = 2

So the solution is x = 2.

5. (2x-1)(2x+1) - 4(x+5)^2 = 19

Expanding and simplifying: 4x^2 - 1 - 4(x^2 + 10x + 25) = 19

Distributing the negative sign: 4x^2 - 1 - 4x^2 - 40x - 100 = 19

Combining like terms: -40x - 101 = 19

Adding 101 to both sides: -40x = 120

Dividing by -40: x = -3

So the solution is x = -3.

6. (6a-1)(6a+1) = 4a(9a+2) - 1

Expanding and simplifying: 36a^2 - 1 = 36a^2 + 8a - 1

Subtracting 36a^2 and -1 from both sides: 0 = 8a

Dividing by 8: a = 0

So the solution is a = 0.

7. (3x-1)^2 - 9x^2 = 10

Expanding and simplifying: 9x^2 - 6x + 1 - 9x^2 = 10

Combining like terms: -6x + 1 = 10

Subtracting 1 from both sides: -6x = 9

Dividing by -6: x = -1.5

So the solution is x = -1.5.

8. (2x+1)^2 - 3(x-5)^2 = (x+3)(x-3)

Expanding and simplifying: 4x^2 + 4x + 1 - 3(x^2 - 10x + 25) = x^2 - 9

Distributing the negative sign: 4x^2 + 4x + 1 - 3x^2 + 30x - 75 = x^2 - 9

Combining like terms: x^2 + 34x - 74 = x^2 - 9

Subtracting x^2 from both sides: 34x - 74 = -9

Adding 74 to both sides: 34x = 65

Dividing by 34: x = 65/34

So the solution is x = 65/34.