Question

Розв'яжіть рівняння: 1) (4х +1)(х-3) = 9;2) (x + 2)(х-3) - (2х -5)(х+3) = x(х-5);3) (6x -5)^2 +(3x-2)(3х + 2) = 36;4) (2x - 1)(4x^2 + 2x+1)-(2x + 5)(4x? - -7) = 41 + x^2.​

Answer 1

Объяснение:

$(4x + 1)( x - 3) \\ x1 = \frac{11 - \sqrt{313} }{8} \\ x2 = \frac{11 + \sqrt{313} }{8}$

Answer 2

1. (4x + 1)(x - 3) = 9

First, expand the left side of the equation:

4x^2 - 12x + x - 3 = 9

Combine like terms:

4x^2 - 11x - 3 = 9

Now, move 9 to the other side of the equation:

4x^2 - 11x - 3 - 9 = 0

Combine constants:

4x^2 - 11x - 12 = 0

Now, factor the quadratic equation:

(4x - 3)(x + 4) = 0

Set each factor equal to zero:

4x - 3 = 0 or x + 4 = 0

Solve for x in each equation:

For 4x - 3 = 0:

4x = 3 x = 3/4

For x + 4 = 0:

x = -4

So, the solutions to the equation are x = 3/4 and x = -4.

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

First, expand each part of the equation:

(x^2 - 3x + 2x - 6) - (2x^2 + 6x - 5x - 15) = x^2 - 5x

Combine like terms:

(x^2 - x - 6) - (2x^2 + x - 15) = x^2 - 5x

Now, distribute the negative sign:

x^2 - x - 6 - 2x^2 - x + 15 = x^2 - 5x

Combine like terms again:

(-x^2 - 2x - 6 + 15) = x^2 - 5x

Simplify the left side:

-x^2 - 2x + 9 = x^2 - 5x

Move all terms to one side of the equation:

0 = x^2 - 5x + x^2 + 2x - 9

Combine like terms:

0 = 2x^2 - 3x - 9

Now, let's solve this quadratic equation. We can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 2, b = -3, and c = -9. Plug these values into the formula:

x = (-(-3) ± √((-3)^2 - 4 * 2 * (-9))) / (2 * 2)

Simplify:

x = (3 ± √(9 + 72)) / 4

x = (3 ± √81) / 4

x = (3 ± 9) / 4

Now, calculate the two possible solutions:

1. x = (3 + 9) / 4 = 12 / 4 = 3
2. x = (3 - 9) / 4 = -6 / 4 = -3/2

So, the solutions to the equation are x = 3 and x = -3/2.

3. (6x - 5)^2 + (3x - 2)(3x + 2) = 36

Let's expand and simplify the left side of the equation:

(6x - 5)^2 + (9x^2 - 4) = 36

Now, expand the squared term:

(36x^2 - 60x + 25) + (9x^2 - 4) = 36

Combine like terms:

36x^2 + 9x^2 - 60x - 4 + 25 = 36

45x^2 - 60x + 21 = 36

Move 36 to the other side of the equation:

45x^2 - 60x + 21 - 36 = 0

Combine constants:

45x^2 - 60x - 15 = 0

Divide the entire equation by 15 to simplify:

3x^2 - 4x - 1 = 0

Now, let's solve this quadratic equation. We can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 3, b = -4, and c = -1. Plug these values into the formula:

x = (-(-4) ± √((-4)^2 - 4 * 3 * (-1))) / (2 * 3)

Simplify:

x = (4 ± √(16 + 12)) / 6

x = (4 ± √28) / 6

Now, simplify the square root:

x = (4 ± 2√7) / 6

Divide the numerator and denominator by 2 to simplify further:

x = (2 ± √7) / 3

So, the solutions to the equation are x = (2 + √7) / 3 and x = (2 - √7) / 3.

4. (2x - 1)(4x^2 + 2x + 1) - (2x + 5)(4x^2 - 7) = 41 + x^2

Let's first expand and simplify each part of the equation:

(2x - 1)(4x^2 + 2x + 1) = 8x^3 + 4x^2 + 2x - 4x^2 - 2x - 1

(2x + 5)(4x^2 - 7) = 8x^3 + 10x^2 - 14x + 20x^2 - 35

Now, subtract the second expression from the first:

(8x^3 + 4x^2 + 2x - 4x^2 - 2x - 1) - (8x^3 + 10x^2 - 14x + 20x^2 - 35) = 41 + x^2

Now, simplify the left side of the equation:

8x^3 + 4x^2 + 2x - 4x^2 - 2x - 1 - 8x^3 - 10x^2 + 14x - 20x^2 + 35 = 41 + x^2

Now, combine like terms on both sides of the equation:

8x^3 - 8x^3 + 4x^2 - 4x^2 + 2x - 2x - 10x^2 + 14x - 20x^2 = 41 + x^2 - 35

Simplify further:

0 = x^2 + 7

Now, subtract 7 from both sides:

x^2 = -7

Take the square root of both sides (remembering to consider both positive and negative roots):

x = ±√(-7)

Since the square root of a negative number results in a complex number, the solutions to this equation are complex numbers:

x = √7i and x = -√7i.