Question

Решить уравнение: Ln(2x^2- 3x-16) = lnx+ln(x+3) -4x^2-x-5=4

Answer 1

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Answer 2

To solve the equation Ln(2x^2 - 3x - 16) = ln(x) + ln(x + 3) - 4x^2 - x - 5 = 4, we need to solve it step by step.

Step 1: Simplify the equation. ln(2x^2 - 3x - 16) = ln(x) + ln(x + 3) - 4x^2 - x - 5 + 4

Step 2: Combine the logarithms on the right side using the logarithm properties. ln(2x^2 - 3x - 16) = ln(x(x + 3)) - 4x^2 - x - 1

Step 3: Apply the logarithm property: ln(a) = ln(b) if and only if a = b. 2x^2 - 3x - 16 = x(x + 3) - (4x^2 + x + 1)

Step 4: Expand the right side. 2x^2 - 3x - 16 = x^2 + 3x - 4x^2 - x - 1

Step 5: Move all terms to one side to set the equation to zero. 2x^2 - 3x - 16 - x^2 - 3x + 4x^2 + x + 1 = 0

Step 6: Combine like terms. 2x^2 - x^2 - 3x + 3x - 16 + 1 = 0

Step 7: Simplify further. x^2 - 15 = 0

Step 8: Now, solve for x by factoring or using the quadratic formula. x^2 - 15 = 0 (x - √15)(x + √15) = 0

x = √15 or x = -√15

So, the solutions to the equation are x = √15 and x = -√15.