Question

5*4^(x-1)-16^(x)+0.25*2^(2x+2)+7=0

Answer 1

Учтём, что 0,25*2^(2x +2) = 1/4*2^2(x+1) = 1/4*4^(x+1)=1/4*4^x*4= 4^x
                    5*4^(x-1) = 5*4^x*4^-1=5/4*4^x
                    16^x = (4^x)²
В нашем уравнении обозначим 4^x = t
получим:
5/4* t - t² + t +7 = 0|*4
- 4t² +5t +4t +28 = 0
-4t² +9t +28 = 0
4t² - 9t -28 = 0 
D = 529
t1=4,    t2= -7/4
а) t = 4                  б) t = -7/4
4^x = 4                      4^x = -7/4   
x = 1                           нет решений
Ответ:1

Answer 2

This is a quadratic equation in terms of 4^x. To solve it, we can use the following steps:

- Rewrite the equation as 5*4^(x-1)-16^(x)+0.25*2^(2x+2)+7=0 - Substitute 4^x with y, so we get 5y/4-16y+0.25y^2+7=0 - Multiply both sides by 4 to get 5y-64y+y^2+28=0 - Rearrange the terms to get y^2-59y+28=0 - Factorize the equation to get (y-56)(y-3)=0 - Solve for y to get y=56 or y=3 - Substitute back 4^x with y, so we get 4^x=56 or 4^x=3 - Take the logarithm of both sides (base 4) to get x=log4(56) or x=log4(3) - Use a calculator to get the approximate values of x as x≈2.48 or x≈0.79

Therefore, the solutions are x≈2.48 or x≈0.79. You can check your answer by plugging these values into the original equation and see if they satisfy it. You can also use an online tool like [Microsoft Math Solver](https://mathsolver.microsoft.com/en/popular-problems) to verify your answer or get more help with similar problems.