Basic functions: Polynomial functions

Theory Root equations

A root equation is an equation in which a root form is present. A common scheme for solving such an equation is as follows:

  1. Isolate the root form.

  2. Take the squares of the left- and right-hand sides and solve the resulting equation.




  3. Check whether the solutions of the squared equation are valid solutions of the original equation

Example

\[x=\sqrt{x+3}-1\]

  1. isolate the root: \(\sqrt{x+3}=x+1\)

  2. Square and solve:
    \(x+3=(x+1)^2=x^2+2x+1\) gives \(x^2+x-2=0\) or \((x+2)(x-1)=0\) and so
    \(x=-2\) or \(x=1\).

  3. Substitution of \(x=-2\) into \(\sqrt{x+3}=x+1\) gives \(1=-1\) and therefore \(x=-2\) is not a valid solution.
    The solution \(x=1\) is valid.
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