Learn Quadratic Functions

A function of the form f(x)= ax2+bx+c, where a,b,c are real numbers and not equal to zero. It is a polynomial of degree '2'. so it can be called as '2nd degree polynomial'.

examples are          1)3x2-6x+4 and

2)-4x2+6x-7   etc.,

If  we draw the graph of aquadratic function  we get a 'Parabola'.A parabola is "U" shaped symmetrical curve. It may be upward or down ward depending upon the sign of 'a', the coefficient of x2,where it is positive or negative.The point where it changes its shape is called "Vertex".or 'Turning point'.

Various Foms of Quadratic function:

1)The form f(x)=ax2+bx+c,is called "General form"

2)The formf(x)=a(x-h)2+k, where h,k are the coordinates of the vertex, is called "Standard form"

3)The form f(x)=a(x-r1)(x-r2),where r1,r2  the roots of the quadratic function , is called "factored form"

Quadratic Equation: If a Quadratic function is made to equal to zero , then it is called "Quadratic Equation"  i.e.,ax2+bx+c=0 is called Quadratic equation.

examples are  1)2x2+3x+4=0 and -9x2+4x-6=0 etc.,

Since it is  apolynomial of 2nd degree it possess two solutions, which are called 'Root' of the equation.

Quadratic Formula: The formula to find out the roots of given equation is

x=(-b+sqrt(b2-4ac))/2a and x=(-b-sqrt(b2-4ac))/2a

the quantity under root is called "Discriminate" .It isdenoted by "Delta"or 'D'.

i.e., D = b2-4ac .

Discriminate gives the nature of the roots .


Nature of Roots - learn quadratic functions


1) If the discriminate is positive , then both the roots are real and distinct.

For the quadrztic  equation with integer coefficiens. if the discriminate is perfect square then the roots are rational numbers, in all other cases they are irrational.

2) If the discriminate is zero , roots are equal and real and they are equal to x=-b/2a

3) if the discrimionate is negative,roots are complex numbers and they are conjugate to each other.

To understand the concept observe the following problems.


Model Problems of learning quadratic functions


Find The  roots and  their nature of the following quadratic equations:

1) x2+2x-3=0

sol: Comparing the given equation with the general form ax2+bx+c=0

a=1;b=2;c=-3

now the disciminant D=2*2-4*1*(-3) = 4+12 = 16, positive so the roots arereal and disticnt

they are x1 = -b+sqrt(D) / 2a = -2 + sqrt(16)/2*1 = 1   and

x2 = -b-sqrt(D) / 2a = -2 - sqrt(16) / 2 * 1 = -3

so the roots are real ,distinct and roots are 1,-3

2) -x2+2x-1 = 0

sol:  comparing it with the standard form  ax2+bx+c = 0, we get

a = -1;b = 2;c = -1

discriminant D = 2*2-4*(-1)*(-1)=4-4=0 so the roots real and equal ,given by

x = -b/2a = -2/2*(-1) =1

hence the roots are real equal and given by 1,1

3) -2x2+2x-2 = 0

Sol: Comparing it with the standard form ax2+bx+c=0,we get

a=-2;b=2;c=-2

D = 2*2-4*(-2)*(-2 ) = 4 -16 = -12 negative  so the roots complex conjugates

x1=-b+sqrt(D)/2a = -2 + sqrt(D) / 2(-2) = (1-2isqrt(3)) / 2

x2=-b-sqrt(D)/2a=-2-sqrt(D)/2(-2)=(1+2isqrt(3))/2 .

In this way we find the roots of the Quadratic equations .