solve the equation 3^2x + 2.3^x - 15 = 0
did you mean ...
3^(2x) + 2(3^x) - 15 = 0 ?
I will assume you did.
let 3^x = t
your equation becomes
t^2 + 2t - 15 = 0
(t+5)(t-3) = 0
t = -5 or t = 3
then 3^x = -5, which has no solution or
3^x = 3
x = 1
To solve the equation 3^(2x) + 2*3^x - 15 = 0, we can use a substitution to simplify it.
Let's substitute y = 3^x. By doing this, the equation becomes a quadratic equation in terms of y.
Therefore, the equation becomes:
y^2 + 2y - 15 = 0
Now, we can solve this quadratic equation by factoring it or by using the quadratic formula.
Factoring method:
We need to find two numbers that multiply to -15 and add up to 2. The numbers that satisfy these conditions are +5 and -3.
So, we can rewrite the equation as:
(y + 5)(y - 3) = 0
Now, we can solve for y:
y + 5 = 0 or y - 3 = 0
For y + 5 = 0, y = -5
For y - 3 = 0, y = 3
Since y = 3^x, we can substitute these values back into the equation:
3^x = -5 (not valid, as an exponent cannot be negative)
3^x = 3
To solve for x, we take the logarithm of both sides of the equation:
log (base 3) of 3^x = log (base 3) of 3
x * log (base 3) of 3 = 1
x * 1 = 1
So, x = 1 is the solution to the equation.