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Ghost Graph - Exploring Extraneous Roots

Extraneous roots are possible when solving radical equations, and are a result of squaring both sides of an equation. Another way of understanding this is that there are "ghosts in the machine", or "spiritum ex machina". If you consider the graph of y=x^2 (a parabola), its inverse graph (y=sqrt(x)) should be a full parabola as well, except one of the arms has been removed in order for it to be a function. This introduces some false solutions (extraneous roots) to the radical equation, since the negative square root is ignored. This graph demonstrates the solution to the radical equation: After solving, there should be two possible solutions: x=1 or x=4 Checking by substitution however, reveals that the x=1 solution does not lead to a valid solution, since it would require to be interpreted as instead. In other words, x=1 is an extraneous root. Click the "Spectral"scope button in the window to show why x=1 is an extraneous root.