Introduction to Quadratic Equations

Quadratic Equations:

The theory of equations deals with the solution of algebraic equations. Quadratic Equations are a particular case of the theory of equations. As it is known that quadratics are represented by parabolas which look similar to a smiley in layman terms. The motion of the balls while juggling, the motion of roller coaster in Disneyland all represents a quadratic. So, quadratics is practically everywhere, be it the orbital motion of the planets, the projectile motion of objects and so on.

General form of a polynomial in x is \(a_n x^n~+~a_{n-1}x^{n-1}~+~⋯..~+~a_1 x~+~a_0\), where \( a_n, a_{n-1}, …..,a_1, a_0\) are constants, \(a_n \) ≠ 0 and n is a whole number. When n = 2 then this polynomial becomes a quadratic expression. Standard form of a quadratic expression is \(ax^2~+~bx~+~c\) ,where a, b and c are constants and a ≠ 0.

The degree of a polynomial is highest power (exponent) of the variable x. The polynomial of degree 2 is known as the quadratic polynomial. So, \(x^2~ +~3x~+~6\) represents a quadratic polynomial of degree 2.

So, a Quadratic is nothing but an expression where the degree of the variable is 2.

When any expression contains an equality sign, it becomes an equation. So, the above expression when written as \(x^2 ~+~3x~+~6\)= 0, it becomes a quadratic equation. The equations like \(x^2\) = 5, \(x^2~+~ 9\) = 0 etc. represent quadratic equations.

Quadratic Equations

Figure 1 Graph of quadratic equation

The graphs of quadratic equations are represented using parabolas which in layman terms can be visualized by looking at a smiley.

So, in order to check whether any equation is quadratic or not, just check whether the degree of the variable is 2 or the highest power which is raised of the variable is 2.


Practise This Question

 A particle is projected from the ground at t = 0 so that on its way it just clears two vertical walls of equal height on the ground. If the particle passes just grazing top of the wall at time  t1 and t2 then calculate

the height of the wall