Question

4x2-4x+33=8√(12+16x-16x2)

Answer 1

Решение приложено

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Answer 2

To solve the equation 4x^2 - 4x + 33 = 8√(12 + 16x - 16x^2), we'll first isolate the radical term on one side and then square both sides of the equation to eliminate the square root. Let's break it down step by step:

1. Start with the given equation: 4x^2 - 4x + 33 = 8√(12 + 16x - 16x^2)

2. Divide both sides of the equation by 8 to simplify it: (1/2)x^2 - (1/2)x + (33/8) = √(12 + 16x - 16x^2)

3. Square both sides of the equation to remove the square root: ((1/2)x^2 - (1/2)x + (33/8))^2 = 12 + 16x - 16x^2

4. Expand the squared term on the left side of the equation: (1/4)x^4 - (1/2)x^3 + (33/16)x^2 - (1/2)x^3 + x^2 - (33/8)x + (33/16)x^2 - (33/8)x + (1089/64) = 12 + 16x - 16x^2

5. Combine like terms on both sides of the equation: (1/4)x^4 - x^3 + (99/16)x^2 - (33/4)x + (1089/64) = 12 + 16x - 16x^2

6. Rearrange the equation to set it equal to zero: (1/4)x^4 - x^3 - (99/16)x^2 + (33/4)x - 16x^2 - 16x + (1089/64) - 12 = 0

7. Simplify further: (1/4)x^4 - x^3 - (177/16)x^2 - (64/4)x + (1089/64) - (768/64) = 0

8. Combine like terms: (1/4)x^4 - x^3 - (177/16)x^2 - (64/4)x + (321/64) = 0

Now, we have a quartic equation. Solving quartic equations can be complex and may not always have simple analytical solutions. The most common way to solve such an equation is through numerical methods or approximation techniques.

If you need an approximate solution, please let me know, and I can use numerical methods to find the roots of the equation.