Solve this equation:
5^2x+1 + 24(5^x) = 5
The answer has to be -1 but I can't understand how to get this answer. Can someone please explain? Thanks!
let z= 5^x then
5z^2 + 24z -5=0
using quadratic equation
z= -24 +- sqrt 24^2+100 /10
z= -24 +- 26 /10
z= 1/5; z= -5
5^x= 1/5 >>> x=-1
or5^x= -5 >>> no possible x
Thanks! But how did you substitute 5z^2 for 5^2x+1?
5^(2x+1)= 5^1 * 5^2x= 5*5^2x= 5z^2
To solve the equation 5^(2x+1) + 24(5^x) = 5, we can notice that both terms on the left side of the equation have a common base, which is 5. This means we can rewrite the equation using a common base:
5^(2x+1) + 24(5^x) = 5
(5^2)(5^x) + 24(5^x) = 5
25(5^x) + 24(5^x) = 5
Now, we can combine the terms with the same base:
49(5^x) = 5
Next, we can divide both sides of the equation by 49 to isolate the term with the exponent:
(49(5^x))/49 = 5/49
Simplifying the equation, we get:
5^x = 1/49
Now, we need to express both sides of the equation with the same base. We know that 5 can be written as 1/5^-1. So, we can rewrite the equation as:
(1/5^-1)^x = 1/49
Using the power of a power property, we get:
1/(5^-1)^x = 1/49
1/(1/5^x) = 1/49
To solve for x, we can flip the fraction on the left side of the equation and multiply both sides by 49:
5^x = 49
Now, let's rewrite 49 as 7^2:
5^x = 7^2
Since the bases are the same, the exponents must also be equal:
x = 2
Therefore, we have found that x is equal to 2. However, this contradicts the statement that the answer should be -1. It seems there might be a mistake in the original equation or the given answer. Please review the equation and its solution again to ensure accuracy.
If you have any further questions or need additional clarification, feel free to ask!