To solve for [tex]\( y \)[/tex] in the equation
[tex]\[
8(2^2)^y = b(2^y) - 1,
\][/tex]
follow these steps:
1. Simplify the Expression: Note that [tex]\( (2^2)^y \)[/tex] simplifies to [tex]\( 2^{2y} \)[/tex]. Therefore, the equation becomes:
[tex]\[
8 \cdot 2^{2y} = b \cdot 2^y - 1.
\][/tex]
2. Substitute: Let [tex]\( z = 2^y \)[/tex]. By substituting [tex]\( z \)[/tex] for [tex]\( 2^y \)[/tex], the equation transforms into:
[tex]\[
8z^2 = bz - 1.
\][/tex]
3. Rearrange in Quadratic Form: Bring all terms to one side to set the equation to zero:
[tex]\[
8z^2 - bz + 1 = 0.
\][/tex]
4. Quadratic Equation: This is a standard quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = 8 \)[/tex], [tex]\( b = -b \)[/tex] (original [tex]\( b \)[/tex]), and [tex]\( c = 1 \)[/tex].
5. Quadratic Formula: The solutions for [tex]\( z \)[/tex] can be found using the quadratic formula:
[tex]\[
z = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A},
\][/tex]
where [tex]\( A = 8 \)[/tex], [tex]\( B = -b \)[/tex], and [tex]\( C = 1 \)[/tex].
However, since we do not have the specific value for [tex]\( b \)[/tex], we cannot proceed with the specific numeric solution. Thus, with the provided information, the final form of this problem is:
[tex]\[
This quadratic equation (8z^2 - bz + 1 = 0) can be solved when the value of \( b \) is provided.
\][/tex]
Upon getting the value of [tex]\( b \)[/tex], you can substitute it into the quadratic formula to find [tex]\( z \)[/tex]. From [tex]\( z = 2^y \)[/tex], you can then solve for [tex]\( y \)[/tex].
