Question

Решите уравнение: (x+2)²/2 - (x²-4/4) - (x-2)²/8 = x²/8

Answer 1

Тримай.
надіюс розбереш

Answer 2

To solve the equation:

(x + 2)²/2 - (x² - 4/4) - (x - 2)²/8 = x²/8

Let's simplify the equation step by step.

Step 1: Expand the squared terms The equation becomes:

(x² + 4x + 4)/2 - (x² - 1) - (x² - 4x + 4)/8 = x²/8

Step 2: Simplify the fractions To simplify the equation, we need to find a common denominator for the fractions. The common denominator is 8.

So, the equation becomes:

(4(x² + 4x + 4))/8 - (8(x² - 1))/8 - (x² - 4x + 4)/8 = x²/8

Simplifying further:

(4x² + 16x + 16)/8 - (8x² - 8)/8 - (x² - 4x + 4)/8 = x²/8

Step 3: Combine like terms Combine the terms with x², x, and constants separately:

(4x² + 16x + 16 - 8x² + 8 - x² + 4x - 4)/8 = x²/8

Simplifying further:

(4x² - 8x² - x² + 16x + 4x + 16 + 8 - 4)/8 = x²/8

Combine like terms:

(-5x² + 20x + 20)/8 = x²/8

Step 4: Move all terms to one side To solve the equation, we need to move all terms to one side of the equation. Let's move all terms to the left side:

-5x² + 20x + 20 - x² = x²

Simplifying further:

-6x² + 20x + 20 = x²

Step 5: Set the equation equal to zero To solve a quadratic equation, we need to set it equal to zero. Let's move all terms to the right side:

-6x² + 20x + 20 - x² - x² = 0

Simplifying further:

-8x² + 20x + 20 = 0

Step 6: Solve the quadratic equation To solve the quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:

The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

For our equation, a = -8, b = 20, and c = 20. Substituting these values into the quadratic formula:

x = (-20 ± √(20² - 4(-8)(20))) / (2(-8))

Simplifying further:

x = (-20 ± √(400 + 640)) / (-16)

x = (-20 ± √(1040)) / (-16)

x = (-20 ± √(16 * 65)) / (-16)

x = (-20 ± 4√65) / (-16)

Simplifying further:

x = (5 ± √65) / 4

So, the solutions to the equation are:

x = (5 + √65) / 4 x = (5 - √65) / 4

These are the two solutions to the given equation.