Question

solution of x² - 1 = 2^x​

Answer 1

Answer:

Step-by-step explanation:

Your math problem

2

−

1

=

2

Yes, there is, however I don't know if its correct or not cause it came to my mind like that when I was sitting and thinking about my math exam. It suddenly came into my mind.

We know that the highest power or order of the equation says the number of solutions that it has. So in this case we must get 2 solutions. But these solutions are in the complex plane.

We can solve the equation as:-

x² + 1 = 0

=> (x+1)² - 2x = 0

=> x+1 = √(2x)

or x - √(2x) + 1 = 0

Now take y=√x

So, the equation changes to

y² - y√2 + 1 = 0

By quadratic formula, we get:-

y = [√2 ± √(2–4)]/2

or √x = (√2 ± i√2)/2 or (1 ± i)/√2 [by cancelling the √2 in numerator and denominator and ‘i' is a imaginary number with value √(-1)]

or x = [(1 ± i)²]/2

So roots are [(1+i)²]/2 and [(1 - i)²]/2

Thus we got two roots but in complex plane. If you put this values in the formula for formation of quadratic equation, that is x²+(a+b)x - ab where a and b are roots of the equation, you will get the equation

x² + 1 = 0 back again

The solution that came to my mind was just another way of solving the same equation. If you solve the root [(1+i)²]/2 you will get ‘i’ and if you solve the other root then you will get ‘-i’ which we have been taught for years that x² + 1 = 0 so x = ±√(-1) or ±i