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Quadratic Equation – Graphical Interpretation of Roots – Complete Guide

Quadratic Equation – Graphical Interpretation of Roots

Quadratic Equation – What are they?

In algebra, quadratic functions are any form of the equation y = axbx + c, where a is not equal to 0, which can be used to solve complex math equations that attempt to evaluate missing factors in the equation by plotting them on a u-shaped figure called a parabola. The graphs of quadratic functions are parabolas; they tend to look like a smile or a frown.

The name Quadratic comes from “quad” meaning square, because the variable gets squared (like  R2).

Let’s move ahead!!

Let R2 be a two dimensional euclidean space.
With usual notation y=f(x) is the function define on R to R.

It means given x in R there is y in R and is associated with calculation of f(x).
Graphically it means for every x in R, compute y and plot the points (x,y), on joining them the sketch of f(x) is observed.

Now let us write the equation as f(x)=0.
It means y=0.

In other words for some x in R there might be y=f(x)=0. Thus the point (x,0) is graphically interpreted as the point of intersection of the function f(x) with the line y=0 (x axis). This point of intersection algebraically referred as the solution or root of the equation f(x)=0.

The number of points of intersection of f(x) and y=0, depends upon the nature of the equation. Equations are broadly classified in two categories:

  • The algebraic equations
  • Transcendental equations.

The polynomial equations of any order is called as algebraic equations, whereas equations involving the terms like exponential, logarithmic, inverse, trigonometric and allegoric are called as transcendental equations. Usually algebraic equations has number of roots equals to the degree of polynomial. Some of these roots may be real and some may be complex. If roots are complex then it appears in complex conjugate also. On the other hand transcendental equations may have infinitely many roots in general.

Here an attempt is made to explain the geometrical meaning of quadratic equations:
(QE: ax2 + bx + c =0). Consider the following three QE

x2 + 4x + 4 = 0 ………..(A)
x2 – 5x + 4 = 0 ……….. (B)
x2 + 2x + 3 = 0 ……….. (C)

By calculus approach one can find discriminant D= ( b2 – 4ac ) to decide the nature of the roots.

Conclusions based on D are as follows;

If D > 0 then QE has two distinct real roots.
If D= 0 then QE has only one real but twice repeated root.
If D < 0 then QE has NO real root.

For QE in (A) D = 0 indicating that the QE has only one real but twice repeated root.
For QE in (B) D = 9 > 0 indicating that the QE has two distinct real roots.
For QE in (C) D = -8 < 0 indicating that the QE has NO real root.

Consider the DESMOS Graphing Calculator generated graph of all the three functions.

The Red graph is for (A), it intersects in only one point (-2,0).

The Sky Blue Red graph is for (B), it intersects in two distinct points (1,0) and (4,0).

The Green graph is for (C), it does NOT intersect with X axis.
Graphical observations are quite consistent with the algebraic findings.

Quadratic Equation
Quadratic Equation

This post was all about Quadratic equations, and the graphic representation of their roots. For a detailed study on various other Statistical & Mathematical blogs which are the basics of your next journey towards Data Science, consider reading the below articles as well:

  1. Study about “ISSUES IN DATA COLLECTION AND STATISTICAL DATA ANALYSIS”, here
  2. Study about “OPERATIONS IN VARIOUS MATHEMATICAL STRUCTURES”, here
  3. Study about “DIFFERENCE BETWEEN THE SYNONYMOUS MATHEMATICAL WORDS IN A TRIPLET”, here

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