Question

4/(2^x+2)-1/(2^x-3)=2. Решить!

Answer 1

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Answer 2

To solve the equation 4/(2^x+2) - 1/(2^x-3) = 2, follow these steps:

Step 1: Find a common denominator for the fractions. The common denominator for the two fractions is (2^x + 2)(2^x - 3).

Step 2: Combine the fractions with the common denominator. 4/(2^x + 2) - 1/(2^x - 3) = (4(2^x - 3) - (2^x + 2))/(2^x + 2)(2^x - 3)

Step 3: Simplify the equation: (4 * 2^x - 12 - 2^x - 2) / (2^x + 2)(2^x - 3) = 2

Step 4: Combine like terms on the left side: (3 * 2^x - 14) / ((2^x + 2)(2^x - 3)) = 2

Step 5: Multiply both sides by the denominator to eliminate the fraction: 3 * 2^x - 14 = 2 * (2^x + 2)(2^x - 3)

Step 6: Expand the right side of the equation: 3 * 2^x - 14 = 4 * (2^x)^2 - 10 * 2^x - 12

Step 7: Rearrange the equation to set it to zero: 4 * (2^x)^2 - 10 * 2^x - (3 * 2^x) - 12 + 14 = 0

Step 8: Combine like terms: 4 * (2^x)^2 - 13 * 2^x + 2 = 0

Step 9: Solve the quadratic equation. Let's use the quadratic formula:

Given a quadratic equation in the form of ax^2 + bx + c = 0, the quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

In our case, a = 4, b = -13, and c = 2:

x = (13 ± √((-13)^2 - 4 * 4 * 2)) / 2 * 4 x = (13 ± √(169 - 32)) / 8 x = (13 ± √137) / 8

So, the two possible solutions are: x = (13 + √137) / 8 x = (13 - √137) / 8

These are the two solutions to the equation.