Quadratic Expressions Study Material for INTERMEDIATE &  IIT JEE

Quadratic Expressions

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Quadratic Expressions very very important  solutions for IPE

Quadratic Expressions  constitute an important part of Algebra. This portion lays the foundation of various important topics and hence it is becomes vital to master the topic. It is a comparatively simple topic and with a bit of hard work it becomes easy to fetch the questions of this topic in the IIT JEE. The various heads covered under this topic include:

• Basic Concepts
• Discriminant of a Quadratic Equation
• Polynomial Equation of Degree n
• The Method of Intervals Wavy Curve Method
• Interval in which the Roots Lie
• Maximum And Minimum Value of a Quadratic Expression
• Resolution of a Quadratic Function Into Linear Factors

 

The word quadratic equation is derived from the Latin word ‘quadratus’ meaning a square. A quadratic equation is any equation having the form
ax²+bx+c =0,
where x represents an unknown, and a, b, and c are constants with a not equal to 0. If a = 0, then the equation is linear, not quadratic. The constants a, b, and c are called the coefficients. While ‘c’ represents the constant term, b is the linear coefficient and ‘a’ the quadratic coefficient. The quadratic equations involve only one unknown and hence are called univariate. The quadratic equations are basically polynomial equations since they contain non-integral powers of x. Since the greatest power is two so they are second degree polynomial equations.

We shall discuss the terms associated with quadratic equations here in brief as they have been discussed in detail in the coming sections.

Discriminant of a ''Quadratic Equations'': The discriminant of a quadratic equation is defined as the number D= b²-4ac and is determined form the coefficients of the equation ax²+bx+c =0. The discriminant reveals the nature of roots an equation has.

Note: b² – 4ac is derived from the quadratic formula

The below table lists the different types of roots associated with the values of determinant.

Discriminant                                         Roots

D < 0                                        two roots which are complex conjugates

D = 0                                        one real root of multiplicity two

D > 0                                        two distinct real roots

D = positive perfect square          two distinct rational roots (assumes a, b and c are rational

Example: Consider the quadratic equation y = 3x²+9x+5. Find its discriminant.

Solution: The given quadratic equation is y = 3x²+9x+5.

The formula of discriminant is D = b²-4ac.

Hence, here a= 3, b= 9 and c= 5 and so the discriminant is given by

D = 9²-4.3.5 = 31.

Polynomial Equation of Degree n:

A polynomial equation is an equation that can be written in the form

axⁿ + bx^n-1 + . . . + rx + s = 0,
where a, b, . . . , r and s are constants. The largest exponent of x appearing in a non-zero term of a polynomial is called the degree of that polynomial.

Examples:

 

1. Consider the equation 3x+1 = 0. The equation has degree 1 as the largest power of x that appears in the equation is 1. Such equations are         called linear equations.
2. x² +x-3 = 0 has degree 2 since this is the largest power of x. such degree 2 equations are called quadratic equations or simply quadratics.
3. Degree 3 equations like x³+2x²-4=0 are called cubic.

A polynomial equation of degree n has n roots, but some of them may be multiple roots. For example, consider x³- 9x²+24x-16 =0.
It is clearly a polynomial of degree 3 and so will have three roots. The equation can be factored as (x-1) (x-4) (x-4) =0. Hence, this implies that the roots of the equation are x=1, x=4, x=4. Hence, the root x=4 is repeated