Solve 5^2x -4(5^x)=12
To solve the equation 5^(2x) - 4(5^x) = 12, we can use a substitution method to simplify the equation. Let's substitute y = 5^x. Then the equation becomes:
5^(2x) - 4y = 12
Now we have a quadratic equation in terms of y, which we can solve using standard techniques. Rearranging the equation, we get:
5^(2x) = 4y + 12
Next, let's substitute y = 5^x back into the equation:
5^(2x) = 4(5^x) + 12
We now have an equation in terms of 5^x. Let's simplify further:
(5^x)^2 = 4(5^x) + 12
Now we have a quadratic equation in terms of 5^x. Let's make a substitution again, this time let's substitute z = 5^x:
z^2 = 4z + 12
Now we have a simpler quadratic equation. Let's rearrange it to the standard quadratic form:
z^2 - 4z - 12 = 0
To solve this equation, we can use factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
z = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 1, b = -4, and c = -12. Plugging in these values:
z = (-(-4) ± √((-4)^2 - 4(1)(-12))) / (2(1))
Simplifying further:
z = (4 ± √(16 + 48)) / 2
z = (4 ± √64) / 2
z = (4 ± 8) / 2
We have two possible solutions for z:
z1 = (4 + 8) / 2 = 12 / 2 = 6
z2 = (4 - 8) / 2 = -4 / 2 = -2
Now that we have the values of z, we can substitute them back into our previous substitution:
For z = 6:
5^x = 6
Taking the logarithm of both sides with base 5:
log5(5^x) = log5(6)
x = log5(6)
For z = -2:
5^x = -2
This equation is not possible because 5^x is always positive, whereas -2 is negative. Therefore, there is no solution for this case.
In conclusion, the solution to the equation 5^(2x) - 4(5^x) = 12 is x = log5(6).