1.1.2—Root transformations

If α is a root of f(x)=0, a new equation with roots such as α+k, kα or 1/α is obtained by substituting the inverse transformation into f, then simplifying.

For a shift y=x−k, replace x by y+k; for a scale y=x/k, replace x by ky. Check whether the question asks for roots or for an equation whose roots are transformed values.

If α satisfies α²−3α+2=0, the roots α+1 satisfy (y−1)²−3(y−1)+2=0, which simplifies to y²−5y+6=0.

Adding k to every root does not mean adding k to every coefficient, and reciprocal roots require multiplying by a suitable power to remove denominators.