Equation Find the number of solutions of equation x-a = square root(x^2-1) if a=0.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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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.`

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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.

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