Zeros Of Quadratic Polynomials

Geometrical interpretation of zeros of quadratic polynomials:

A polynomial of degree 2 is called a quadratic polynomial. For any quadratic polynomial , $ax^{2}+bx+c$ , $a\neq 0$

the graph of the corresponding equation $ax^{2}+bx+c$ has one of the two shapes either open upwards like U or open downwards like , ∩ depending on whether a > 0 or a < 0. (These curves are called parabolas.)

The zeroes of a quadratic polynomial $ax^{2}+bx+c$ , $a\neq 0$ are precisely the x-coordinates of the points where the parabola representing $y=ax^{2}+bx+c$ intersects the x-axis.

Now the following three cases can happen.

Case-(i) The graph cuts x-axis at two distinct points. (See fig 1.)


Fig. 1

In this case , we get two distinct solutions of y.

Case(ii) The graph cuts the x-axis at exactly one point (see fig.2)

fig.(2)

The x-coordinate of A is the only zero for the quadratic polynomial in this case.

Case(iii) The graph is either completely above the x-axis or completely below the x-axis. (see Fig. 3).

fig(3)

So, the quadratic polynomial has no zero in this case.

Thus we see geometrically that a quadratic polynomial can have either two distinct zeroes or two equal zeroes (i.e., one zero), or no zero. This also means that a polynomial of degree 2 has atmost two zeroes.