If q and w are the roots of the equation2x2−px+7=0

Question

Norah Small · Accepted Answer

The following result (and its generalizations to higher degree polynomials) is very useful. Consider the equation

a x^{2} + b x + c = 0

, where

a \neq 0

. Then the sum of the roots is

- \frac{b}{a}

and the product of the roots is

\frac{c}{a}

. In our case, we have

q + w = \frac{p}{2}

and

q w = \frac{7}{2}

We will find a quadratic equation whose roots are

\frac{q}{w}

and

\frac{w}{q}

without calculating

\frac{q}{w}

or

\frac{w}{q}

. Note that

\frac{q}{w} + \frac{w}{q} = \frac{q^{2} + w^{2}}{q w} = \frac{{(q + w)}^{2} - 2 q w}{q w} = \frac{{(q + w)}^{2}}{q w} - 2

Substitute our known values of

q + w

and

q w

. We get

\frac{q}{w} + \frac{w}{q} = \frac{p^{2}}{14} - 2 = \frac{p^{2} - 28}{14}

Note that trivially the product of

\frac{q}{w}

and

\frac{w}{q}

is 1

x^{2} - (\frac{p^{2} - 28}{14}) x + 1 = 0

If we wish, we can change to the equivalent

14 x^{2} + (28 - p^{2}) x + 14 = 0

If q and w are the roots of the equation 2x^2-px+7=0