Determine x from the expression:sqrt(x^2+2)=x+3

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remove the radical, hence, you need to square it, such that:

`(sqrt(x^2 + 2)) = (x + 3)^2 => x^2 + 2 = x^2 + 6x + 9`

Reducing duplicate terms yields:

`2 = 6x + 9 => 2 - 9 = 6x => -7 = 6x => x = -7/6`

You need to test the value `x = -7/6` in equation such that:

`sqrt((-7/6)^2 + 2) = -7/6 + 3 => sqrt((49 + 72)/36) = (-7 + 18)/6`

`sqrt(121/36) = 11/6 => 11/6 = 11/6`

Hence, evaluating the solution to the given equation, yields `x = -7/6.`

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll start by imposing constraints of existence of the square root.

x^2 + 2 > 0

Since te value of x^2 is always positive, no matter the value of x is, the expression x^2 + 2 > 0.

Now, we'll square raise both sides to get rid of the square root.:

[sqrt(x^2 + 2)]^2 = (x+3)^2

We'll expand the square from the right side:

x^2 + 2 = x^2 + 6x + 9

We'll subtract x^2 + 6x + 9 and we'll eliminate like terms:

-6x - 9 + 2 = 0

-6x - 7 = 0

We'll add 7 both sides:

-6x = 7

We'll divide by -6:

x = -7/6

x = -1.1(6)

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