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State the investor cash flows
Contributions of 100 occur at time zero and year one. Final liquidation pays 242 at year two. Contributions are investor outflows and liquidation is an inflow. All intervals are one year.
Solve in terms of the growth factor
Let x = 1+r > 0. Multiplying the PV equation by x² gives 100x²+100x−242 = 0. The positive root is [−100+√106800]/200. The other root is negative and falls outside this positive accumulation-factor convention.
The transformation preserves the original annual cash-flow equation.
Reconstruct the final amount
The positive x gives r = 13.401346%. At year two, the original 100 grows to 100x², while the later 100 grows to 100x. Their sum is 242. The later contribution receives only one interval of growth.
Keep the root problem specific
This original schedule has one positive-domain root. That observation does not prove every investment schedule has a unique IRR. A different pattern of inflows and outflows can change the number or relevance of roots, and irregular dates need an explicit time convention.