Question

1) (x – 3)(x + 7) - (x + 7)(x – 8) = 0; 2) (4x -9)(x - 2) + (1 - x)(x - 2) = 0;3) 0,2x (x – 5) + 8(x – 5) = 0;4) 7(x – 7) - (x – 7)2 = 0.​

Answer 1

Ответ:

1) х=-7

2) х₁=2; х₂=$\frac{8}{3}$

3) х₁=-40; х₂=5

4) х=7

Пошаговое объяснение:

1)(x – 3)(x + 7) - (x + 7)(x – 8) = 0

(x+7)×(x-3-(x-8)) = 0

(x+7)×(-3+8) = 0

(x+7)×5 = 0

5х+35=0

5х=-7

х=-7

2) (4x -9)(x - 2) + (1 - x)(x - 2) = 0

(х-2)×(3х-8)=0

х-2=0         3х-8=0

х₁=2;           х₂=$\frac{8}{3}$

Answer 2

Let's solve each of these quadratic equations step by step:

1. (x - 3)(x + 7) - (x + 7)(x - 8) = 0

First, expand both sets of parentheses:

(x^2 - 3x + 7x - 21) - (x^2 - 8x + 7x - 56) = 0

Now, simplify the equation:

(x^2 + 4x - 21) - (x^2 - x - 56) = 0

Now, distribute the negative sign:

x^2 + 4x - 21 - x^2 + x + 56 = 0

Now, combine like terms:

5x + 35 = 0

Subtract 35 from both sides:

5x = -35

Divide by 5:

x = -7

So, the solution to the equation is x = -7.

2. (4x - 9)(x - 2) + (1 - x)(x - 2) = 0

First, expand both sets of parentheses:

(4x^2 - 8x - 9x + 18) + (x - x^2 - 2x + 2) = 0

Now, simplify the equation:

(4x^2 - 17x + 18) + (x - x^2 - 2x + 2) = 0

Now, combine like terms:

4x^2 - 17x + 18 + x - x^2 - 2x + 2 = 0

Now, combine like terms again:

3x^2 - 18x + 20 = 0

Divide the entire equation by 3 to simplify:

x^2 - 6x + 20/3 = 0

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

x = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 1, b = -6, and c = 20/3. Plug these values into the quadratic formula:

x = (-(-6) ± √((-6)² - 4(1)(20/3))) / (2(1))

x = (6 ± √(36 - 80/3)) / 2

x = (6 ± √(108/3 - 80/3)) / 2

x = (6 ± √(28/3)) / 2

x = (6 ± √(28)/√(3)) / 2

x = (6 ± 2√(7)/√(3)) / 2

Now, simplify further by dividing both the numerator and denominator by 2:

x = (3 ± √(7)/√(3))

Rationalize the denominator by multiplying the numerator and denominator by √(3):

x = (3 ± √(7)√(3)/3)

So, the solutions to the equation are:

x = (3 + √(21))/3 x = (3 - √(21))/3

3. 0.2x(x - 5) + 8(x - 5) = 0

First, distribute the constants and variables:

0.2x^2 - 1x + 8x - 40 = 0

Now, combine like terms:

0.2x^2 + 7x - 40 = 0

To solve this quadratic equation, you can multiply the entire equation by 5 to eliminate the decimal:

5(0.2x^2 + 7x - 40) = 5(0)

x^2 + 35x - 200 = 0

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

x = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 1, b = 35, and c = -200. Plug these values into the quadratic formula:

x = (-35 ± √(35² - 4(1)(-200))) / (2(1))

x = (-35 ± √(1225 + 800)) / 2

x = (-35 ± √2025) / 2

x = (-35 ± 45) / 2

Now, we have two possible solutions:

x = (-35 + 45) / 2 = 10 / 2 = 5

x = (-35 - 45) / 2 = -80 / 2 = -40

So, the solutions to the equation are:

x = 5 and x = -40.

4. 7(x - 7) - (x - 7)^2 = 0

Let's simplify the equation:

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

Now, distribute the negative sign:

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

Combine like terms:

6x - x^2 - 98 = 0

Now, rearrange the terms:

-x^2 + 6x - 98 = 0

Multiply the entire equation by -1 to make it easier to work with:

x^2 - 6x + 98 = 0

This is a quadratic equation in standard form. To solve it, you can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 1, b = -6, and c = 98. Plug these values into the quadratic formula:

x = (-(-6) ± √((-6)² - 4(1)(98))) / (2(1))

x = (6 ± √(36 - 392)) / 2

x = (6 ± √(-356)) / 2

Since the term inside the square root is negative, this equation has no real solutions. It has complex solutions:

x = (6 ± √(356)i) / 2

x = (3 ± √(356)i)

So, the solutions are:

x = 3 + √(356)i and x = 3 - √(356)i.

These are the solutions to the given equations.