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Suppose an object is thrown upward with initial throw velocity $V_i$ and object's velocity become zero ($V_f=0$) as it arrives at a height $h$ from the throw point. The motion is along y-axis and the upward direction is taken positive. Then we have following equation to calculate Vi.

$$-2gh=V_f^2 - V_i^2$$

Which implies,

$$V_i = \pm \sqrt{2gh+V_f^2}$$

Now we have 2 values for initial velocity and both satisfies the equation however only positive value is the true one because initial velocity got to be in the upward direction which we assumed positive. So what is the negative value telling us considering that it is not an extraneous root?

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As you said, the negative "extra" root has no physical meaning. This happens when dealing with powers or absolute values. It's really important to check if your solution is logical or not, as you did it.

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This equation of motion you are using i.e. $V_f^2 - V_i^2 = 2as$ is a scalar equation. Hence the solution, $$|V_i| = \sqrt{V_f^2 - 2as}$$ tells that the absolute value of initial velocity i.e. initial speed is $\sqrt{V_f^2 - 2as}$ . Since speed is rate of change of distance with time, and distance cannot be -ve as it is the physical length covered by an object in its motion, hence the speed is also +ve.

So by this equation, you only get the initial speed(scalar). To get the velocity(speed + its direction), you should use the other two equations of motion which are vector equations

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