The sum and the product of the roots of a quadratic can be read off from its coefficients without solving the equation: if has roots and , then and . These relations are Vieta’s formulas; the solutions on this page and later ones cite them as “Vieta”.
For example, : the roots are .
Problem 9.2.1. Let have roots and .
(a) Show that for all .
(b) Find in terms of and .
(c) Find in terms of and .
Show solution
(a), and
cannot be , so cancel it:
so and . Since ,
Alternatively: .
(b).
(c).
Problem 9.2.2. Find the sum and the product of the roots of each equation: and .
Show solution
Well, and , so we have:
and .
and .
Problem 9.2.3. Let be the roots of .
(a) Relate ,,, to and .
Show solution
(a) and , so and ; also and (for nonzero roots). Then
So , the same as (that is, at ). Likewise
Adding,
Next, in terms of . Since
and . So we want in terms of and . Trying products:
Squaring the sum: , so , and
Problem 9.2.4. Find if and are the roots of .
Show solution
So , and
With :
Real parts: , so or (check: ✓, ✓).
Imaginary parts: .
So and .
Problem 9.2.5. Given that the sum of the zeros, the product of the zeros and the sum of the coefficients of are equal, show that all three of these quantities must be equal to .
Show solution
, with
Then
so .
Example 9.2.6. Roots , of , one of which is .
Show solution
. Here
With :
Also tried: , so .
Exercises
Exercise 9.2.7., where are the roots of . Find .
Show solution
By Vieta, and .
Then .
Substituting for (and using ):
Now , and from ,
so , and .
Exercise 9.2.8. The roots of satisfy ,; and for , and . Solve for .
Show solution
For fun:
so the two equations are and .
Exercise 9.2.9.Show solution
The roots satisfy , i.e. , so . So and : indeed and .
By the quadratic formula,
where
so , i.e. or .
Answer:,.
Exercise 9.2.10. If has solutions for such that each solution is the negative of the other, find in terms of and .
Show solution
We want . By Vieta,
Show solution
We want for the roots in , with ; find in terms of .
Also , i.e. ,.
If the roots are , the quadratic is a multiple of , with no term: it should look like , i.e. .
Expanding,
so the -term is , and its coefficient must be . By Vieta,
gives , so , i.e. , and
Also , so .
Exercise 9.2.11. Find all for which the zeros of differ by .
Show solution
(So , i.e. .) Then
and . Doubling,
and with :
(Equivalently, and give , so , i.e. .)