$P(x) =3x^2-11x-2$
$P(x)=x^2 -x -11$
$P(x)=x^2 + x -897$

Graphing quadratics Polynomial

Lets us assume
y= p(x) where p(x) is the polynomial of quadratic type
Now we can plot the equation y=p(x) on the Cartesian plane by taking various values of x and y obtained by putting the values. The plot or graph obtained can be of any shapes
The zero's of the polynomial are the points where the graph meet x axis in the Cartesian plane. If the graph does not meet x axis ,then the polynomial does not have any zero's.
Let us take some useful polynomial and shapes obtained on the Cartesian plane

what is a quadratic equation

When we equate Quadratic Polynomial with zero, it is called Quadratic equation
$ax^2 +bx+c =0$ where a ≠0

Examples:

$6x^2-x-2=0$
$x^2 -x -20=0$
$x^2 + x -300 = 0$

It is used in many field and it has many application in Mathematics

For example
John has a area of a rectangular plot is 528 m^{2}.The length of the plot (in meters) is one more than twice its breadth.
We need to find the length and breadth of the plot
For solution , we can assume breadth is x ,then lenght would be (2x+1)
Now $528=x(2x+1)$
or $2x^2+x -528 =0$
Which is a quadratic equation and the required representation of the problem mathematically

Solution or root of the Quadratic equation
A real number $\alpha$ is called the root or solution of the quadratic equation if
$a \alpha^2 +b \alpha +c=0$

Some other points to remember

The root of the quadratic equation is the zeroes of the polynomial p(x).

We know from chapter two that a polynomial of degree can have max two zeroes. So a quadratic equation can have maximum two roots

A quadratic equation has no real roots if b^{2}- 4ac < 0

A graphing quadratic equation is same as graphing quadratic polynomial as explained above