Tiffany from Island School and Andrei from Tudor Vianu National
College, Romania sent in very good solutions.
(1) If you keep p constant and change q, the graph of
y=x2 + px + q (1)
is translated parallel to the y axis. Consequently the number of
intersections of the graph with the x axis (i.e. of real solutions
of the equation x2+px+q=0) changes. This number is related to the
discriminant of the equation: D = p2 -4q (2).
- if D > 0 there are 2 real, distinct solutions
- if D = 0 there is a repeated (or double) real solution
- if D < 0 there are no real solutions.
a) For p=-5, q=-6: D = 49 and the x-intercepts are (-1,0)
and (6,0) showing 2 real solutions to the equation .
b) For p = -5, q = 4: D = 9 and the x-intercepts are (1,0)
and (4,0) showing 2 real solutions.
c) For p = -5, q = 7: D = -3 and there are no intersection
with the x axis and no real solutions.
(2) If you keep q constant and change p the intersection of the
graph of y=x2 + px + q with the y-axis is kept fixed at the point
(0,q). The number of intersections with the x-axis varies and it
could be 0, 1 or 2 depending again on D.
From relation (2) it
is easy to observe that D has the same value if p changes
sign and that for p2 > 4q and for p2 < -4q there are 2
distinct real solutions, for p2 = 4q there are two equal real
solutions and for the rest there are no real solutions.
a) For p=-10, q = 16 the solutions are x=2 and x=8.
b) For p=-8, q=16 the graph is tangent to the x-axis and there is
a repeated real solution x=4.
c) p = -6, q=16 there are no x-intercepts and we shall see that
there are 2 complex conjugate solutions.
(3) and (4)
When the point (p,q) is in the region below the parabola p2 = 4q, showing D = p2 - 4q > 0, the graph of y = x2 + px + q crosses the x-axis in two distinct points and the equation x2 + px + q = 0 has 2 distinct real solutions. In this case the the
roots show up on the u-axis in the Argand Diagram (called the
real axis).
When the point (p,q) enters the region above the parabola p2 = 4q the graph of y = x2 + px + q no longer crosses the x-axis and
the equation x2 + px + q = 0 has no real solutions. In this
region the movement of the point (p,q) in the red frame leaves a
red track showing D = p2 - 4q < 0.
When the point (p,q) is on the boundary between the two regions,
that is on the parabola p2 = 4q, the graph of y = x2 + px + q
is tangent to the x-axis, there are 2 equal real solutions shown by
two coincident points on the u-axis in the Argand diagram.
The roots show up on the v-axis in the Argand Diagram (called the
imaginary axis) when p=0 and q > 0.
(5) The roots of the equation x2 -6x +13 = 0 are the complex
numbers: z1 = 3 + 2i and z2 = 3 - 2i where
i =
__ Ö-1
.
They satisfy the Viéte relations z1 + z2 = 6 and z1 × z2 = 13. Checking that z12 -6z1 + 13 = 0 and z22 -6z2 + 13 = 0 we verify that these are solutions of the equation.
(7) Two complex roots of the quadratic equation x2 + px +q = 0
(where p and q are real numbers and p2 < 4q) are
z1 =
-p +
______ Öp2 - 4q
2
= u1 + iv1 and z2 =
-p -
______ Öp2 - 4q
2
= u2 + iv2.
We see that
u1 = u2 =
-p2
and
v1 = - v2 =
_______ Öp2 - 4q
so the points in the Argand diagram representing these
solutions are reflections of each other in the real axis.
The sum of the roots is z1 + z2 = p.
The product of the roots is