Question

1) x÷(3+x) + 4x+6÷(x^2+4x+3)=0 2) a) 2x+1/(x-3) ≤ 1 b) (x-2)(x+5)÷(x-1)^2

Answer 1

x^3 - 3x^2 - 4x = 0,

x(x^2-3x-4)=0,

x1=0,

x^2-3x-4=0,

x2=-1, x3=4;

 

(2x - 1)^4 - x^2 = 0,

((2x-1)^2)^2-x^2=0,

((2x-1)^2-x)((2x-1)^2+x)=0,

(2x-1)^2-x=0,

4x^2-5x+1=0,

D=9,

x1=1/4, x2=1;

(2x-1)^2+x=0,

4x^2-3x+1=0,

D=-7<0; нет решений

 

x^3 - 3x^2 - x +3 = 0,

x^2(x-3)-(x-3)=0,

(x-3)(x^2-1)=0,

x-3=0, x1=3;

x^2-1=0, x^2=1, x2=-1, x3=1

Answer 2

1. To solve the equation x÷(3+x) + 4x+6÷(x^2+4x+3) = 0, we can follow these steps:

First, let's simplify the equation by finding common denominators:

x÷(3+x) + 4x+6÷(x^2+4x+3) = 0

Multiply the first fraction by (x^2+4x+3) and the second fraction by (3+x):

x(x^2+4x+3) + (4x+6)(3+x) = 0

Now, expand and simplify:

x^3 + 4x^2 + 3x + 12x + 6x + 18 = 0

Combine like terms:

x^3 + 4x^2 + 21x + 18 = 0

Next, let's factor the equation:

(x + 2)(x + 3)(x + 3) = 0

Now, we have three possible solutions:

x + 2 = 0 -> x = -2 x + 3 = 0 -> x = -3 x + 3 = 0 -> x = -3

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

2. a) To solve the inequality 2x + 1/(x - 3) ≤ 1, we can follow these steps:

Multiply the entire inequality by (x - 3) to remove the denominator:

2x(x - 3) + 1 ≤ (x - 3)

Expand and simplify:

2x^2 - 6x + 1 ≤ x - 3

Rearrange the inequality to have all terms on one side:

2x^2 - 6x - x + 1 + 3 ≤ 0

2x^2 - 7x + 4 ≤ 0

Now, we need to find the values of x that satisfy this inequality. To do that, we can factor the quadratic equation:

(2x - 1)(x - 4) ≤ 0

Now, we have two critical points: x = 1/2 and x = 4. We can use these points to create intervals and determine the sign of the expression within each interval.

Testing the interval x < 1/2:

Choose a value in the interval, let's say x = 0:

(2(0) - 1)(0 - 4) ≤ 0 (-1)(-4) ≤ 0 4 ≤ 0

This is false.

Testing the interval 1/2 ≤ x < 4:

Choose a value in the interval, let's say x = 2:

(2(2) - 1)(2 - 4) ≤ 0 (3)(-2) ≤ 0 -6 ≤ 0

This is true.

Testing the interval x ≥ 4:

Choose a value in the interval, let's say x = 5:

(2(5) - 1)(5 - 4) ≤ 0 (9)(1) ≤ 0 9 ≤ 0

This is false.

Therefore, the solution to the inequality 2x + 1/(x - 3) ≤ 1 is 1/2 ≤ x < 4.

b) To simplify the expression (x - 2)(x + 5) ÷ (x - 1)^2, we can follow these steps:

Start by factoring the numerator:

(x - 2)(x +