Question

Решить уравнения: 1) (2х^2+х+2)\(4х^2+5х-14)=(2х^2+х+6)\(4х^2+5х-10) 2) (х-2)(х-3)(х-4)(х-5)=24

Answer 1

2) (x-2)*(x-3)*(x-4)*(x-5) = 24
x^2-7x+16=0
x1+x2=7
x1*x2=16
x1=1
x2=6
1)(8*(x^2+2x-8))/((4x^2+5x-14)(4x^2+5x-10))=0
x^2+2x-8=0
X1X2=(-2 +корінь з 36)/2
1/(4x^2+5x-14)=0
1/(4x^2+5x-10)=0
x=-4
x=2

Answer 2

To solve the equation:

(2х^2+х+2)/(4х^2+5х-14) = (2х^2+х+6)/(4х^2+5х-10)

We can start by simplifying both sides by finding a common denominator, which is (4x^2 + 5x - 14)(4x^2 + 5x - 10).

Multiplying both sides by this common denominator gives:

(2x^2 + x + 2)(4x^2 + 5x - 10) = (2x^2 + x + 6)(4x^2 + 5x - 14)

Expanding and simplifying, we get:

8x^4 - 14x^3 - 35x^2 + 20x + 40 = 8x^4 - 14x^3 - 35x^2 + 26x + 84

Subtracting 8x^4 - 14x^3 - 35x^2 from both sides, we get:

20x + 40 = 26x + 84

Subtracting 26x from both sides and simplifying, we get:

-6x = 44

Dividing both sides by -6, we get:

x = -22/3

Therefore, the solution to the equation is x = -22/3.

To solve the equation:

(x - 2)(x - 3)(x - 4)(x - 5) = 24

We can start by finding the factors of 24, which are 1, 2, 3, 4, 6, 8, 12, and 24. We can then try different combinations of these factors to see which one satisfies the equation.

One possible combination is:

(x - 2)(x - 3)(x - 4)(x - 5) = 8 x 3

Expanding both sides and simplifying, we get:

x^4 - 14x^3 + 61x^2 - 90x + 24 = 24

Subtracting 24 from both sides and simplifying, we get:

x^4 - 14x^3 + 61x^2 - 90x = 0

Factoring out x, we get:

x(x^3 - 14x^2 + 61x - 90) = 0

We can see that x = 0 is one solution. To find the other solutions, we can use synthetic division or the rational root theorem to factor x^3 - 14x^2 + 61x - 90.

Using synthetic division with x = 2, we get:

2 | 1 - 14 + 61 - 90 | 2 - 24 + 74 | 1 - 12 + 37 - 16

Therefore, x^3 - 14x^2 + 61x - 90 = (x - 2)(x^2 - 12x + 37)

Using the quadratic formula to solve for the roots of x^2 - 12x + 37, we get:

x = (12 ± sqrt(12^2 - 4(1)(37))) / 2(1)

x = 6 ± sqrt(11)

Therefore, the solutions to the equation are x = 0, x = 2, x = 6 + sqrt(11), and x = 6 - sqrt(11).