If the equations x^2-p x+q=0, x^2-r x+s=0 have a common root...

If the equations $x^{2}-p x+q=0, x^{2}-r x+s=0$ have a common root which is the harmonic mean between their other two roots, prove that
(i) $(\mathrm{q}-\mathrm{s})^{2}=(\mathrm{p}-\mathrm{r})(\mathrm{qr}-\mathrm{ps})$
(ii) $p s(3 q+s)=q r(3 s+q)$

Key Concepts

Algebraic Manipulation and Symmetry

The process of bridging given conditions, like the harmonic mean property, with the algebraic structure provided by Vieta's formulas typically involves careful algebraic manipulation. By expressing the sums and products of the roots in symmetric forms, one can derive identities that reveal deep relationships among the coefficients. Exploiting symmetry in the equations not only simplifies the calculations but also illustrates underlying invariant relationships that hold true under the given conditions.

Common Root Condition

When two quadratic equations share a common root, this imposes a constraint linking the coefficients of the equations. Recognizing and utilizing the fact that one of the roots appears in both equations is key to establishing relationships between the coefficients. This insight allows one to eliminate variables and to derive identities that must hold between the parameters of the equations, which is often seen in the resulting algebraic identities.

Vieta's Formulas

Vieta's formulas form a cornerstone in understanding the relationship between the roots of a polynomial equation and its coefficients. In the context of quadratic equations, they relate the sum and the product of the roots to the coefficients of the quadratic. This principle is particularly useful when we have multiple quadratic equations with overlapping roots, as it allows us to set up equations linking the parameters of these quadratics to their roots, and ultimately to each other.

Harmonic Mean

The harmonic mean is a type of average that is defined as the reciprocal of the arithmetic mean of the reciprocals of the values. In problems that involve roots of equations, stating that one root is the harmonic mean of the other two introduces an additional relationship among the roots. This relation is instrumental in setting up an equation that joins the reciprocals of the non?common roots with the common root, thereby linking it back to the coefficients via Vieta's formulas.

Please give Ace some feedback