To solve the equation 2^(x^2 - 35) = 4^x, you can convert the right side of the equation to have the same base as the left side. Since both 2 and 4 are powers of 2, we can rewrite 4 as (2^2):
2^(x^2 - 35) = (2^2)^x
Using the properties of exponents, we can simplify the equation further:
2^(x^2 - 35) = 2^(2x)
Now that the bases are the same, we can equate the exponents:
x^2 - 35 = 2x
Next, rearrange the equation to have all the terms on one side:
x^2 - 2x - 35 = 0
Now we have a quadratic equation in terms of x. To solve this, we can factorize or use the quadratic formula. Let's factorize it:
(x-7)(x+5) = 0
By applying the zero product property, we can set each factor equal to zero:
x - 7 = 0 --> x = 7
or
x + 5 = 0 --> x = -5
So the potential solutions for x are x = 7 and x = -5.