How to reduce exponential equation to quadratic equation form?

Question

I was teaching a class and came across a question related to quadratic equations. Question was about reducing given exponential equation to quadratic form and solve by factorization later. I would have solved that if bases were same or I could manipulate any of side in term of power of other. But it was impossible to write one side in term of power of other.

 

Now I' like to know if there is any method to reduce such equations to quadratic ones.?

Answer 1

An example of that is this:

 

Solve for:

 

e^2x - 4e^x + 3 = 0;

 

You can actually manipulate the power but it would be like this: (we know that (b^m)ⁿ = b^mn right?)

 

So we can rewrite the equation as:

 

(e^x)² - 4e^x + 3 = 0

 

Now, we let the e^x = m (or any other variable that you like except x) and it will be:

 

m² - 4m + 3 = 0

 

Voila, your quadratic equation. By factoring:

 

(m-3)(m-1) = 0

 

m - 3 = 0          m - 1 = 0

 

m = 3               m = 1

 

Return the value of 'm':

 

e^x = 3               e^x = 1

 

x = ln3              x  = ln1 = 0

 

*The actual equation given here is y = e^2x - 4e^x + 3. But since solving for x also means solving for the "zeroes" of an equation, these are actually the x-intercepts. Meaning, you will let y = 0 and proceed on the process above.*