Math problem: It is known that 
and 
are the roots of the quadratic equation
Derive a quadratic equation whose roots are 
and 
.
Discussion and hints: Let us consider a general quadratic equation
Without loss of generality we may assume that 
To derive the formulas for the roots of the general quadratic equation (1) we complete the square on the left hand side to get
Introduce the discriminant 
of (1). It is clear from the formula above that if 
the quadratic equation (1) has two real roots, which are given by the quadratic formula:
If 
then (1) admits one (double) real root
If 
then (1) has two complex-conjugate roots
From the formulas that we have derived above, it is easy to compute (Viète's formulas)
Figure 1
Let us also say a few words about maximum and minimum of the
quadratic polynomial function
If
then the polynomial
has no maximum value, but has the minimum of
at
If
then the polynomial
ha no minimum, but it has the maximum value of
at
The graph of the function
is a parabola. Its vertex is at the point
If
the parabola is concave up, if
the parabola is concave down. The vertical axis of such a parabola is the line
See
Figure 1 for more details.
To complete our discussion let us consider the inequality defined by a quadratic polynomial:
If
and
(
) the inequality (3) holds true for any value of
(see
Figure 1). If
and
(
) the inequality (3) is impossible for any value of
) (see
Figure 1). If
and
(
) the inequality (3) holds true for all values of
satisfying
or
, where
are the roots of the quadratic equation (1) (see
Figure 1). If
and
(
) the inequality (3) holds true for all values of
satisfying
, where
are the roots of the quadratic equation (1) (see
Figure 1). If If
and
then the inequality (3) holds true for all values of If
and
(except
, see
Figure 1). If If
and
then the inequality (3) is impossible for any value of
and
(see
Figure 1).
Solution
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