Equation Find the number of solutions of equation x-a = square root(x^2-1) if a=0.
You need to solve for x the following irrational equation, such that:
`x - a = sqrt(x^2 - 1)`
The problem provides the information that `a = 0` , such that:
`x = sqrt(x^2 - 1) => x^2 = ( sqrt(x^2 - 1) )^2`
`x^2 = x^2 - 1`
Reducing duplicate terms `x^2` yields:
`0 = -1 ` invalid
Hence, evaluating the number `n` of solutions to the given equation, under the given conditions, yields that `n = 0.`
For the beginning we'll impose the constraint of existence of the square root:
x^2 - 1 >=0
The expression x^2 - 1 is positive over the intervals (- infinite , -1] U [1 ; +infinite).
Now, we'll put a = 0 and we'll solve the given equation:
x = sqrt(x^2 - 1)
We'll raise to square both sides, in order to eliminate the square root:
x^2 = x^2 - 1
We'll subtract x^2:
x^2 - x^2 = -1
0 = -1 not true
The equation has no solutions for a = 0.