Question

(x − 1)(x − 2) + (x + 3)(x − 3) + Зx + 18 = 0 (Зx − 5)2 − 5(5 − 6x) = 0 (x − 7)2 + 7(2x − 14) = 0

Answer 1

a) x^2 - 2x - x + 2 + x^2 - 9 + 3x + 18 = 0

2x^2 + 0 +11 =0

б ) 6х - 10 - 25 + 30x =0

36x - 35 = 0

36x = 35

x = 35/36

Answer 2

To solve each of these equations, we will simplify and manipulate them to find the values of 'x' that satisfy the equations.

1. (x − 1)(x − 2) + (x + 3)(x − 3) + 3x + 18 = 0

Let's start by expanding and simplifying the equation:

(x − 1)(x − 2) + (x + 3)(x − 3) + 3x + 18 = 0 (x^2 - 3x + 2) + (x^2 - 3x - 9) + 3x + 18 = 0 x^2 - 3x + 2 + x^2 - 3x - 9 + 3x + 18 = 0 2x^2 - 3x + 11 = 0

Now, we have a quadratic equation in standard form: ax^2 + bx + c = 0, where a = 2, b = -3, and c = 11.

To solve this quadratic equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Substituting the values, we get:

x = (3 ± √((-3)^2 - 4211)) / 2*2 x = (3 ± √(9 - 88)) / 4 x = (3 ± √(-79)) / 4

Since the discriminant (√(b^2 - 4ac)) is negative, there are no real solutions for 'x' in this equation.

2. (Зx − 5)^2 − 5(5 − 6x) = 0

Let's simplify the equation step by step:

(3x - 5)^2 - 5(5 - 6x) = 0

Now, expand the square term:

(9x^2 - 30x + 25) - 5(5 - 6x) = 0

Distribute the -5 on the second term:

9x^2 - 30x + 25 - 25 + 30x = 0

Combine like terms:

9x^2 = 0

Now, divide by 9 to solve for 'x':

x^2 = 0 x = ±√0 x = 0

So, the only real solution for 'x' in this equation is x = 0.

3. (x − 7)^2 + 7(2x − 14) = 0

Let's simplify the equation:

(x - 7)^2 + 7(2x - 14) = 0

Expand the square term:

(x^2 - 14x + 49) + 7(2x - 14) = 0

Now, distribute the 7:

x^2 - 14x + 49 + 14x - 98 = 0

Combine like terms:

x^2 - 49 = 0

Add 49 to both sides:

x^2 = 49

Now, take the square root of both sides:

x = ±√49

x = ±7

So, the two real solutions for 'x' in this equation are x = 7 and x = -7.

To summarize the solutions:

1. (x − 1)(x − 2) + (x + 3)(x − 3) + 3x + 18 = 0 has no real solutions.
2. (Зx − 5)^2 − 5(5 − 6x) = 0 has one real solution: x = 0.
3. (x − 7)^2 + 7(2x − 14) = 0 has two real solutions: x = 7 and x = -7.