Conditions for square root equationsWhy do i have solve square root equations considering conditions and what are the conditions? Please, could you give a solved problem?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll take the equationsquare root (x-1) =1 – square root(2-x).

Since it is impossible for a square root to have a negative radicand, we'll impose that the radicand to be positive, or zero.

We'll impose the constraints of existence of square roots:

x - 1>=0

x >= 1

2 -x >=0

x =<2

The interval of admissible values for x is [1 ; 2].

Now, we'll solve the equation. We'll subtract sqrt(2-x) both sides:

sqrt(x-1) = 1 - sqrt(2-x)

We'll raise to square both sides:

x - 1 = 1 - 2sqrt(2-x)+ 2 - x

We'll combine like terms:

x - 1 = 3  -x - 2sqrt(2-x)

We'll subtract 3 - x both sides:

2x - 4 = -2sqrt(2-x)

We'll divide by -2:

2 - x = sqrt(2-x)

We'll raise to square both sides:

4 - 4x + x^2 = 2 - x

We'll subtract 2 - x:

x^2 - 3x + 2 = 0

(x-1)(x-2) = 0

x-1=0

x = 1

x-2=0

x = 2

The values of the roots have to be located in the interval of admissible values for x.

Since both values belong to the interval of admissible values, we'll accept them as real solutions of the equation.

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