Question

3^(x)-3^(2-x)-8=0....

Answer 1

3^х - 3^(2 - х) - 8 = 0
3^х - 9/3^х - 8 = 0
3^х = t > 0
t - 9/t - 8 = 0
(t^2 - 8t - 9)/t = 0
t^2 - 8t - 9 = 0, при условии t <> 0
По теореме Виета:
t1 = -1, t2 = 9
3^х = 9
х = 2

Answer 2

To solve the equation 3^x - 3^(2 - x) - 8 = 0, we can use some algebraic techniques to simplify and find the value of x.

Let's first rewrite the equation as follows: 3^x - 3^(2 - x) = 8

Now, we can observe that 3^x and 3^(2 - x) have the same base, which is 3. We can use this fact to simplify the equation.

Step 1: Convert 3^(2 - x) to a term with the same base (3^x): We know that 3^a * 3^b = 3^(a + b). So, 3^(2 - x) can be written as 3^2 * 3^(-x) = 9 * 3^(-x).

Now, the equation becomes: 3^x - 9 * 3^(-x) = 8

Step 2: Substitute a variable to simplify the equation: Let's substitute y = 3^x. Now, the equation becomes: y - 9/y = 8

Step 3: Transform the equation into a quadratic equation: To do this, we'll multiply the whole equation by y to eliminate the fraction: y^2 - 9 = 8y

Step 4: Set the equation to zero and factorize: y^2 - 8y - 9 = 0

Step 5: Solve for y using quadratic formula or factoring: Using the quadratic formula, y = [8 ± √(8^2 - 4 * 1 * (-9))] / 2 y = [8 ± √(64 + 36)] / 2 y = [8 ± √100] / 2 y = [8 ± 10] / 2

So, we have two potential solutions for y:

1. y = (8 + 10) / 2 = 18 / 2 = 9
2. y = (8 - 10) / 2 = -2 / 2 = -1

Step 6: Find the corresponding values for x: Now that we have the values of y, we can find the corresponding values of x using the substitution y = 3^x.

For y = 9: 3^x = 9 Take the logarithm base 3 of both sides: x = log₃(9) x = 2

For y = -1: 3^x = -1 (There is no real solution for this since 3^x is always positive for real values of x.)

So, the real solution to the equation 3^x - 3^(2 - x) - 8 = 0 is x = 2.