Video Lecture & Short Notes : QUADRATIC EQUATION

Equation of the form   is known as quadratic equation in one variable and forms a second degree polynomial, where;

• a , b, c are the real numbers
• a ≠ 0

Consider an equation ax2+bx+c=0 then we can further define

Complete quadratic equation: where both the coefficients and the constant term are non-zero. a≠ 0, b≠ 0, c≠ 0.

Example

2x2+ 3x+ 4= 0

Pure quadratic equation: where except the coefficient of x2, others can be zero.  a≠ 0 , b=0, c=0

Examples

2x2= 0

2x2+3=0

It is the value of x for which the quadratic equation becomes zero.

A quadratic equation (the two degree polynomial) will have at most 2 zeros and hence at most 2 roots.

NOTE: Zeros and roots of a second degree polynomial are the same.

If the equation is given by ax2+bx+c, thus if       then the roots of quadratic equation are given by the quadratic formula given as,

METHODS TO SOLVE A QUADRATIC EQUATIONS

• Factorizing

By Splitting the mid-term

Example:  Consider an equation x2 + 5x + 6= 0

This equation can be written as

x2+3x+2x+6= 0

x(x+3)+2(x+3)= 0

(x+2)(x+3)= 0

Hence the roots are x= -2 or x= -3

• Completing the squares

The equation is converted in the form a2+b2 or a2-b2 by subtracting or adding a suitable constant.

Example: Consider an equation 4x2 – 2x – 5 = 0

This equation can be written as

While solving further we want the x2 term by itself, therefore we have to divide the whole equation by whatever is multiplied by it.

In this case divide the whole equation by 4 to get

Now, we are going to work with the x term along with its sign, here take    as the numeric coefficient here.

Multiply   to this numeric coefficient to get the derived coefficient

Square the derived coefficient to get

Get back to the equation

Add   to both sides of the equation

By doing this we have created the left hand side which is a perfect square and hence can be converted into a squared binomial.

The above equation is in complete square form.

To get the roots, apply square root to both the sides.

Hence, the roots of the equation are as follows

Find the determinant and use the quadratic formula to take out the roots.

Consider the above example itself 4x2 – 2x – 5 = 0

The quadratic formula, as mentioned previously is:

• Discriminant method

1. Find
 VALUE OF D ROOTS NATURE OF ROOTS D>0 –b+ /2a  and –b- /2a Real and distinct D=0 -b/2a Real and equal OR coincidental D<0 no real roots Not Real (imaginary roots)

Sum and product of roots

Let α and β be the two distinct roots of the equation , then

• Sum of roots is given as

= α + β =

• Product of roots is given as

= α.β =

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