Roots And Coefficients
In a quadratic equation, the relation between its roots and coefficients is not negligible. In lower secondary, knowing how to use and to apply the Viete Theorem is more than enough.
The Viete Theorem states that if are the real roots of the equation , then:
Proof: (need not know)
By the factor theorem, the equation has roots iff
Expanding the R.H.S. gives us , thus by the comparing the coefficients yields us the theorem.
There is also an inverse theorem, basically it states that for any 2 real numbers , the equation
has as its two real roots.
The inverse theorem basically says that the Viete theorem works vice versa.
Viete theorem is useful in the sense that if you are given the values of in a quadratic equation, not only the roots can be solved (by using simultaneous equations), many expressions in can be given by , provided the expression is a function of and .
For example, the following expressions are often used and solved by Viete theorem:
By using these expressions, it is possible to establish new equations to solve for .
Given that is a real number not less than , such that the equation in :
has two distinct real roots .
- If , find the value of .
- Find the maximum value of .
The equation has 2 distinct real roots implies that , thus
Therefore, we obtain .
By Viete Theorem, and , thus
Therefore , i.e. . By applying the quadratic formula, we obtain
By completing the squares,
, since .
Thus the maximum value of is .
Extended Viete Formula
If and are the roots of the cubic equation , then
, , .
If and are the roots of the quartic equation , then
It is known that the roots of the equation
are all integers. How many distinct roots does the equation have?
Sum of roots = .
Product of roots = .
Since is a prime, we can conclude from the simultaneous equations that:
Therefore, there are 3 distinct roots.
Roots and Discriminants
Roots are the solutions to a quadratic equation while the discriminant is a number that can be calculated from any quadratic equation.
Discriminant of a quadratic equation = =
Nature of the solutions :
has at least a real root, find the range of m.
i) When , then , so there is a real root .
ii) When , then , so there is a real root .
iii) When , then
which implies that and .
Thus, the range of is .