Question

E) (x − 2)² + 48 = (2 – 3x)² g) -2x² + 5x + 6 = 4x² + 5x d) (x + 1)² = (2x − 1)² f) 7x + 1 = 3x² - 2x + 1 h) (2x − 3)² – (3x - 2)² = 20(x-1)​

Answer 1

Ответ:

e) Уравнение имеет два корня:

$x = \frac{1 + 2 \sqrt{6} }{2}$

$x = \frac{1 - 2 \sqrt{6} }{2}$

g) Уравнение имеет два корня:

$x = - 2$

$x = 0$

f) Уравнение имеет два корня:

$x = 3$

$x = 0$

h) Уравнение имеет два корня:

$x = 1$

$x = - 5$

Пошаговое объяснение:

Подробное решение уравнений на снимке:

Answer 2

Let's solve these equations one by one:

E) (x − 2)² + 48 = (2 – 3x)²

First, expand both sides:

(x^2 - 4x + 4) + 48 = (4 - 12x + 9x^2)

Now, simplify and rearrange:

x^2 - 4x + 52 = 9x^2 - 12x + 4

Bring all terms to one side:

0 = 9x^2 - x^2 - 12x + 4x - 4 - 52

0 = 8x^2 - 8x - 56

Divide the entire equation by 8 to simplify:

0 = x^2 - x - 7

Now, let's solve for x. You can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

In this equation, a = 1, b = -1, and c = -7:

x = (-(-1) ± √((-1)² - 4(1)(-7))) / (2(1))

x = (1 ± √(1 + 28)) / 2

x = (1 ± √29) / 2

So, the solutions are:

x = (1 + √29) / 2 x = (1 - √29) / 2

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G) -2x² + 5x + 6 = 4x² + 5x

First, move all terms to one side to set the equation to zero:

-2x² + 5x + 6 - 4x² - 5x = 0

Combine like terms:

-6x² + 6 = 0

Now, divide both sides by -6 to simplify:

x² - 1 = 0

Now, solve for x:

x² = 1

Take the square root of both sides:

x = ±√1

So, the solutions are:

x = 1 x = -1

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D) (x + 1)² = (2x − 1)²

First, expand both sides:

x² + 2x + 1 = 4x² - 4x + 1

Now, move all terms to one side:

0 = 4x² - x² - 4x - 2x + 1 - 1

0 = 3x² - 6x

Now, factor out 3x:

3x(x - 2) = 0

Now, set each factor equal to zero:

3x = 0 or x - 2 = 0

For 3x = 0, divide by 3:

x = 0

For x - 2 = 0, add 2 to both sides:

x = 2

So, the solutions are:

x = 0 x = 2

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F) 7x + 1 = 3x² - 2x + 1

First, move all terms to one side to set the equation to zero:

3x² - 2x + 1 - 7x - 1 = 0

Combine like terms:

3x² - 9x = 0

Now, factor out 3x:

3x(x - 3) = 0

Now, set each factor equal to zero:

3x = 0 or x - 3 = 0

For 3x = 0, divide by 3:

x = 0

For x - 3 = 0, add 3 to both sides:

x = 3

So, the solutions are:

x = 0 x = 3

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H) (2x − 3)² – (3x - 2)² = 20(x-1)

First, expand both sides:

(4x² - 12x + 9) - (9x² - 12x + 4) = 20x - 20

Now, simplify and rearrange:

4x² - 12x + 9 - 9x² + 12x - 4 = 20x - 20

Now, combine like terms:

-5x² + 5 = 20x - 20

Move all terms to one side:

-5x² - 20x + 5 + 20 = 0

Simplify:

-5x² - 20x + 25 = 0

Divide the entire equation by -5 to simplify:

x² + 4x - 5 = 0

Now, let's solve for x. You can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

In this equation, a = 1, b = 4, and c = -5:

x = (-4 ± √(4² - 4(1)(-5))) / (2(1))

x = (-4 ± √(16 + 20)) / 2

x = (-4 ± √36) / 2

x = (-4 ± 6) / 2

So, the solutions are:

x = (-4 + 6) / 2 = 2 / 2 = 1 x = (-4 - 6) / 2 = -10 / 2 = -5

The solutions are:

x = 1 x = -5