Answered

Looking for trustworthy answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.

Solve for [tex]\( x \)[/tex]:
[tex]\[ 2^x + 2^{x-1} - 48 = 0 \][/tex]

Sagot :

GinnyAnswer
To solve the equation [tex]\( 2^x + 2^{x-1} - 48 = 0 \)[/tex], let's follow these logical steps.

1. Rewrite the Equation:

Notice that [tex]\( 2^{x-1} \)[/tex] can be written as [tex]\( \frac{2^x}{2} \)[/tex]. Therefore, the given equation can be rewritten as:
[tex]\[ 2^x + \frac{2^x}{2} - 48 = 0 \][/tex]

2. Combine Like Terms:

Combine the terms involving [tex]\( 2^x \)[/tex]. We express [tex]\( 2^x + \frac{2^x}{2} \)[/tex] with a common factor:
[tex]\[ 2^x + \frac{2^x}{2} = 2^x \left(1 + \frac{1}{2}\right) = 2^x \left(\frac{3}{2}\right) \][/tex]
Hence, the equation becomes:
[tex]\[ \frac{3}{2} \cdot 2^x - 48 = 0 \][/tex]

3. Isolate the Exponential Expression:

To isolate [tex]\( 2^x \)[/tex], multiply both sides of the equation by [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[ 2^x = 48 \cdot \frac{2}{3} \][/tex]
Simplifying the right-hand side:
[tex]\[ 2^x = 32 \][/tex]

4. Solve for [tex]\( x \)[/tex]:

We know that [tex]\( 32 \)[/tex] can be expressed as a power of [tex]\( 2 \)[/tex]. Specifically:
[tex]\[ 32 = 2^5 \][/tex]
Thus, we have:
[tex]\[ 2^x = 2^5 \][/tex]

5. Equating the Exponents:

Since the bases are the same, we can equate the exponents:
[tex]\[ x = 5 \][/tex]

Therefore, the solution to the equation [tex]\( 2^x + 2^{x-1} - 48 = 0 \)[/tex] is:
[tex]\[ x = 5 \][/tex]
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.