In the equation `2^(2y+3)=65(2^(y)-1)+57` then, solutions are

Options

A. `4,-3`
B. `-3,-3`
C. `3,-3`
D. `3,3`

Answer

To solve the equation \( 2^{2y+3} = 65(2^y - 1) + 57 \), we will follow these steps:

### Step 1: Simplify the equation
Start by simplifying the right-hand side of the equation:

\[ 2^{2y+3} = 65(2^y - 1) + 57 \]

Expanding the right-hand side:

\[ 2^{2y+3} = 65 \cdot 2^y - 65 + 57 \]

Combine the constants:

\[ 2^{2y+3} = 65 \cdot 2^y - 8 \]

### Step 2: Rewrite the left-hand side
Using the property of exponents, we can rewrite \( 2^{2y+3} \) as:

\[ 2^{2y+3} = 2^3 \cdot (2^y)^2 = 8(2^y)^2 \]

So now our equation looks like:

\[ 8(2^y)^2 = 65 \cdot 2^y - 8 \]

### Step 3: Rearrange the equation
Rearranging gives us:

\[ 8(2^y)^2 - 65(2^y) + 8 = 0 \]

### Step 4: Substitute \( t = 2^y \)
Let \( t = 2^y \). The equation now becomes:

\[ 8t^2 - 65t + 8 = 0 \]

### Step 5: Solve the quadratic equation
To solve the quadratic equation \( 8t^2 - 65t + 8 = 0 \), we can use the quadratic formula:

\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 8 \), \( b = -65 \), and \( c = 8 \).

Calculating the discriminant:

\[ b^2 - 4ac = (-65)^2 - 4 \cdot 8 \cdot 8 = 4225 - 256 = 3969 \]

Now applying the quadratic formula:

\[ t = \frac{65 \pm \sqrt{3969}}{16} \]

Calculating \( \sqrt{3969} = 63 \):

\[ t = \frac{65 \pm 63}{16} \]

This gives us two solutions:

1. \( t = \frac{128}{16} = 8 \)
2. \( t = \frac{2}{16} = \frac{1}{8} \)

### Step 6: Back substitute for \( y \)
Now we substitute back for \( t = 2^y \):

1. For \( t = 8 \):
\[ 2^y = 8 \implies y = 3 \]

2. For \( t = \frac{1}{8} \):
\[ 2^y = \frac{1}{8} \implies 2^y = 2^{-3} \implies y = -3 \]

### Final Solutions
Thus, the solutions for the equation \( 2^{2y+3} = 65(2^y - 1) + 57 \) are:

\[ y = 3 \quad \text{and} \quad y = -3 \]