It is believed that Babylonians were the first to solve quadratic equations. Greek mathematician Euclid developed a geometrical approach for finding solutions of quadratic equations.
Solving of quadratic equations, in general form, is often credited to ancient Indian mathematicians.
Any equation of the form p(x) = 0, where p(x) is a polynomial of degree 2, is a quadratic equation.
But when we write the terms of p(x) in descending order of their degrees, then we get the standard form of the equation.
That is, ax2 + bx + c = 0, a ≠ 0 is called the standard form of a quadratic equation.
Convert Statement to Quadratic Equation
John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.
Solution: Let john has x marbles, then Jivanti has 45-x marbles, since total number of marbles is 45.
Now both of them lost 5 marbles each, thus now marble count is
- John : x-5
- Jivanti : 45-x -5 or 40-x
Given that product of new marble count is 124, that is Marbles with john * Marbles with Jivanti = 124.
(x-5) * (40-x) = 124
Or x2 – 45x + 324 = 0 , where x is the count of marbles with John.
Numerical 1 : Check if x(x + 1) + 9 = (x + 2) (x – 2) is a quadratic equation
Solution: On expanding the equation, we get
x2 + x + 9 = x2 + 2x -2x -4
x + 13 = 0.
Since, it is not of the form ax2 + bx + c = 0, it is not a quadratic equation.
Solution of a Quadratic Equation by Factorization
For a quadratic equation ax2 + bx + c = 0, a ≠ 0
if aα2 + bα + c = 0. We also say that x = α is a solution of the quadratic equation
if aβ2 + bβ + c = 0. We also say that x = αis a solution of the quadratic equation.
Thus if α & β are solution of the quadratic equation ax2 + bx + c = 0. Then we can write
ax2 + bx + c = 0 = (x- α) (x-β), where α & β are solution of this quadratic equation.
In case of factorization method, we re-write the quadratic equation in the form (x- α) (x-β).
Example: Find solution for quadratic equation x2 – 5x + 6 =0
By splitting the middle term we can rewrite the equation as
x2 – 3x -2x + 6 =0 as (-3x * -2x = x2 * 6)
Or x(x-3) -2 (x-3) 0
Or (x-3)(x-2) = 0
Or x=3 or x= 2 are solutions for this quadratic equation.
Solution of a Quadratic Equation by Completing the Square
In this case, we try to reduce a quadratic equation in the form (px+q)2 = n2
This x = (+n –q)/p or (-n –q)/p
For Example, lets solve quadratic equation x2 + 4x – 5=0
We can rewrite this equation as
(x2 + 4x + 4) -4 – 5=0
or (x+2)2 = 32
or X = +3 -2 or -3-2
or x =1 or -5
Nature of Roots
A quadratic equation ax2 + bx + c = 0 has
(i) two distinct real roots, if b2 – 4ac > 0,
(ii) two equal real roots, if b2 – 4ac = 0,
(iii) no real roots, if b2 – 4ac < 0.