Related Equations

A quadratic equation    ax² + bx +c    has roots α & β so    (x – α)(x - β)=0    implies that
α + β = - b//a     and  αβ  = c/a.

related equation (x – α’)(x - β’) = 0 has a different combination of {a,b,c} in its coefficients.

Alpha and beta are remarkably difficult to reproduce in iWeb, and often fail to translate across browsers. This page uses α as alpha and x as “ex” ; so now you know what your browser has done with the intended characters].       I attempted to improve this behaviour on 20130313. The rewrite in 2017 moved many such characters (but not all) to unicode versions, which ought (theory) to work on all browsers; it is testing such a statement that is, quite simply, too much effort. If you see fault in reproduction of characters, do please tell me. I am well awware that spaces don’t transfer well - just look at the way the [marks] fail to line up. Oh, for a tabbing function.
                                                                                                                                   DJS       


Example: If 6x² - 11x - 7 has roots α & β find the related equation with roots 1/α and 1/β.

From the given equation,  the sum of the roots is α + β = 11/6,   and the product is   αβ = 7/6
For the new equation  α’ + β’ = 1/ α + 1/β  =  (α+β) / αβ =11/7       and     α’β’ = 1/αβ = 6/7
The new equation needs the sum of the roots, α’+ β’, now seen to be 11/7, and the product, α’’β’, seen to be 6/7.  Rewrite the equation with integer coefficients.
So the new equation is      7x
² – 11x + 6 =0; this special case swaps the coefficients.


Exercise:

1     6x² – 11x +1 = 0;                                                          new eqn α² , β²

2    6x² – 11x +1 = 0;                                                           new eqn α³ , β³

3    5x² – 13x +7 = 0;                                                           new eqn α⁻² , β⁻²

4    2x² + 19 x - 6 = 0;                                                         new eqn α² + β² , α² - β²

5    3x² –  5 x + 6 = 0;                                                          new eqn α + 2β, β+2α

6    6x² – 26x + 24 = 0                                                         new eqn αβ, 1/αβ

7    4x² + 7x - 36 = 0;                                                          new eqn  β-α, α-β

8    x³ + 4x² + 7x -+ 6 = 0;                                                  new eqn 1/α, 1/β. 1/

9    4x³ + 4x² + 7x - 3 = 0;                                                  new eqn α+β, β+∂, ∂+α

10    2x³ + 4x² + 27x - 36 = 0;                                             new eqn α² - β² , 2αβ


Syllabus quote (2013?) probably Edexcel Further Pure Maths, a Y11 course that year.

Polynomials and Rational functions

Recall and use the relations between the roots and coefficients of polynomial equations of degree 2, 3 and 4 only

Use a given simple substitution to obtain an equation whose roots are related in a simple way to those of the original equation

Sketch graphs of simple rational functions, including the determination of oblique asymptotes, in cases where the degree of the numerator and denominator are at most 2 (detailed plots not required, but sketches are expected to show significant features, such as turning points, asymptotes and intersections with the axes)


See also the page Roots of Equations, adjacent.

© David Scoins 2017