Outline:
What does Quadratic mean? Oops, not 2. It means square. So, a polynomial of degree 2, when equated to 0 gives Quadratic equation. Customarily, it is represented as 
, a≠0 where a, b and c are constants.
Wait! Why a≠0?
If we place a=0, it becomes 
. This is linear, not quadratic.
How to find x:
We have a nice and cozy formula which pumps out x for every possible quadratic equation. Quadratic formula:
![\[x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\]](https://i0.wp.com/tutortonight.com/wp-content/ql-cache/quicklatex.com-a1fb31fe6af1cd9133714b814b96c93c_l3.png?resize=156%2C40&ssl=1 "Rendered by QuickLaTeX.com")
Why +- ?
That’s because we have 2 solutions.
In fact, there is a rule: Total solutions of a polynomial equation are equal to the degree of polynomial. Whether those roots are real or complex, that’s an entirely different topic.
Graphical representation of quadratic equation: A quadratic equation of the form 
represents a parabola.
Remember parabola? That looks like a dish antenna, or an arch bridge.
If a>0, it opens up.
If a<0, it opens down.
The points where the curve intersects x axis would be the “roots” or “solution” of that quadratic equation.
Example:
If we have an equation with a>0, It will be a parabola – opening up
Since it intersects the x-axis at two distinct points, it has 2 real roots.
Key points of a parabola:
Vertex: In simple words, the turning point is called Vertex of quadratic equation. Consequently, this will be the same point where Local extrema occur.
Axis of symmetry: This term is self-explanatory. The axis about which the curve is symmetric. In above graphs, it is y-axis (x=0).
x-intercepts: The points where the curve intersects x-axis. These are also the roots of the quadratic equation.
y-intercepts: The points where the curve intersects y-axis.
Application: Shape of parabola is very useful in positioning headlights, parabolic receptors, satellites, tracing the ball which you throw from a height, engineering of arch bridges and so on.
Do check out this video .I have covered various questions on this topic.
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