Join the IDNLearn.com community and get your questions answered by experts. Join our interactive Q&A community and access a wealth of reliable answers to your most pressing questions.
Sagot :
Certainly! Let's solve the equation step-by-step:
Given the equation:
[tex]\[ 5^{x^2 + 2x + 7} = 125^{2x + 1} \][/tex]
### Step 1: Simplify the Bases
First, we know that [tex]\(125\)[/tex] can be expressed as a power of [tex]\(5\)[/tex]:
[tex]\[ 125 = 5^3 \][/tex]
Thus, we can rewrite the given equation as:
[tex]\[ 5^{x^2 + 2x + 7} = (5^3)^{2x + 1} \][/tex]
### Step 2: Simplify the Exponents
Using the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex], we can rewrite the right side:
[tex]\[ (5^3)^{2x + 1} = 5^{3(2x + 1)} = 5^{6x + 3} \][/tex]
Now the equation becomes:
[tex]\[ 5^{x^2 + 2x + 7} = 5^{6x + 3} \][/tex]
### Step 3: Set the Exponents Equal
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[ x^2 + 2x + 7 = 6x + 3 \][/tex]
### Step 4: Rearrange and Simplify the Equation
Rearrange the equation to bring all the terms to one side:
[tex]\[ x^2 + 2x + 7 - 6x - 3 = 0 \][/tex]
Combine like terms:
[tex]\[ x^2 - 4x + 4 = 0 \][/tex]
### Step 5: Solve the Quadratic Equation
Now we have a quadratic equation:
[tex]\[ x^2 - 4x + 4 = 0 \][/tex]
We can solve this quadratic equation using various methods such as factoring, completing the square, or using the quadratic formula. In this case, the equation factors nicely:
[tex]\[ (x - 2)^2 = 0 \][/tex]
So, we find:
[tex]\[ x - 2 = 0 \][/tex]
Thus:
[tex]\[ x = 2 \][/tex]
### Conclusion
By solving the equation, we find:
[tex]\[ x = 2 \][/tex]
Therefore, the solution to the equation [tex]\(5^{x^2 + 2x + 7} = 125^{2x + 1}\)[/tex] is:
[tex]\[ x = 2 \][/tex]
Given the equation:
[tex]\[ 5^{x^2 + 2x + 7} = 125^{2x + 1} \][/tex]
### Step 1: Simplify the Bases
First, we know that [tex]\(125\)[/tex] can be expressed as a power of [tex]\(5\)[/tex]:
[tex]\[ 125 = 5^3 \][/tex]
Thus, we can rewrite the given equation as:
[tex]\[ 5^{x^2 + 2x + 7} = (5^3)^{2x + 1} \][/tex]
### Step 2: Simplify the Exponents
Using the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex], we can rewrite the right side:
[tex]\[ (5^3)^{2x + 1} = 5^{3(2x + 1)} = 5^{6x + 3} \][/tex]
Now the equation becomes:
[tex]\[ 5^{x^2 + 2x + 7} = 5^{6x + 3} \][/tex]
### Step 3: Set the Exponents Equal
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[ x^2 + 2x + 7 = 6x + 3 \][/tex]
### Step 4: Rearrange and Simplify the Equation
Rearrange the equation to bring all the terms to one side:
[tex]\[ x^2 + 2x + 7 - 6x - 3 = 0 \][/tex]
Combine like terms:
[tex]\[ x^2 - 4x + 4 = 0 \][/tex]
### Step 5: Solve the Quadratic Equation
Now we have a quadratic equation:
[tex]\[ x^2 - 4x + 4 = 0 \][/tex]
We can solve this quadratic equation using various methods such as factoring, completing the square, or using the quadratic formula. In this case, the equation factors nicely:
[tex]\[ (x - 2)^2 = 0 \][/tex]
So, we find:
[tex]\[ x - 2 = 0 \][/tex]
Thus:
[tex]\[ x = 2 \][/tex]
### Conclusion
By solving the equation, we find:
[tex]\[ x = 2 \][/tex]
Therefore, the solution to the equation [tex]\(5^{x^2 + 2x + 7} = 125^{2x + 1}\)[/tex] is:
[tex]\[ x = 2 \][/tex]
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com has the solutions you’re looking for. Thanks for visiting, and see you next time for more reliable information.