Question
upstudy study bank question image url

Solve for \( x \). \[ 27^{5 x+8}=3^{x^{2}+6 x+38} \] If there is more than one solution, separat

Ask by Bird Knight. in Saudi Arabia
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

الحلان للاختصار هما \( x=2 \) و \( x=7 \).

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Extra Insights

To solve the equation \[ 27^{5x + 8} = 3^{x^2 + 6x + 38}, \] we first express 27 as a power of 3: \[ 27 = 3^3. \] Then, we can rewrite \( 27^{5x + 8} \) as: \[ (3^3)^{5x + 8} = 3^{3(5x + 8)} = 3^{15x + 24}. \] Now we can rewrite the original equation: \[ 3^{15x + 24} = 3^{x^2 + 6x + 38}. \] Since the bases are the same, we equate the exponents: \[ 15x + 24 = x^2 + 6x + 38. \] Next, we rearrange the equation to set it to zero: \[ x^2 + 6x + 38 - 15x - 24 = 0, \] which simplifies to: \[ x^2 - 9x + 14 = 0. \] Now, we can factor the quadratic: \[ (x - 7)(x - 2) = 0. \] Setting each factor to zero gives us: \[ x - 7 = 0 \quad \Rightarrow \quad x = 7, \] \[ x - 2 = 0 \quad \Rightarrow \quad x = 2. \] Thus, the solutions to the equation are: \[ \boxed{7} \text{ and } \boxed{2}. \]

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy