Quadratic Functions:
Quadratic functions involve the second power of the variable but no higher power. The kinetic energy of a moving body is given by K.E. =1/2mv^2 , where m represents the mass and v represents the velocity of the object.
In general, A quadratic function is function that may be defined by the equation y=ax^2+bx+c, where a, b, and c are complex numbers constants or parameters, but a≠0. Notice that the restriction a≠0 assures that y=ax^2+bx+c defines a non-linear function.
The following equations define the quadratic functions:
1. y=x^2
2. y=-4x^2
3 y=x^2-6x+8
Quadratic Equations:
An equation in which the highest power of the variable i.e., degree is two is called a quadratic equation.
ax^2+bx+c=0, where a,b,c are constants, is a general quadratic equation.
Examples:
1. 5x^2+2x=0
2. x^2+2x-1=0
3. 58x^2+52x +6=0
4. 9x^2+6x-1=0
Consider the quadratic equations ax^2+bx+c=0, where a, b, c € R and a≠0
Roots of the equation is given by x=(-b±√(b^2-4ac))/2a .
What is plus/minus sign?
The plus/minus sign is used because every quadratic equation has maximum two solution.
Nature of Roots:
The nature of the roots, real or imaginary, depends on the b^2-4ac.
This is generally denoted by D also called Discriminant.
1. If D=0, Roots are real and equal.
2. If D>0, Roots are real and distinct.
3. If D<0, Roots are imaginary.