9th class jee quadratic expression
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Quadratic Equations 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 Equation: 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.
You may refer the Sample Papers to get an idea about the types of questions asked.
Polynomial Equation of Degree n
A polynomial equation is an equation that can be written in the form
axn + bxn-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:
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.
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.
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.
Descarte’s Rule of Signs
This is a very famous rule that helps in getting an idea about the roots of a polynomial equation. The rule states that the number of positive real roots of Pn(x) = 0 cannot be more than the number of sign changes. Similarly, the number of negative roots cannot be more than the number of sign changes in Pn(-x).
For instance, consider the equation P5(x) = x5+2x4-x3+x2-x+2 = 0.
Now as stated above, since this equation has four sign changes so it can have at most four positive real roots. Now if we see P5(-x), it has only one sign change as
P5(-x) = -x5+2x4+x3+x2+x+2 = 0
Hence, it can have only one negative real root.
Please mark it as brainliest answer.
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 Equation: 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.
You may refer the Sample Papers to get an idea about the types of questions asked.
Polynomial Equation of Degree n
A polynomial equation is an equation that can be written in the form
axn + bxn-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:
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.
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.
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.
Descarte’s Rule of Signs
This is a very famous rule that helps in getting an idea about the roots of a polynomial equation. The rule states that the number of positive real roots of Pn(x) = 0 cannot be more than the number of sign changes. Similarly, the number of negative roots cannot be more than the number of sign changes in Pn(-x).
For instance, consider the equation P5(x) = x5+2x4-x3+x2-x+2 = 0.
Now as stated above, since this equation has four sign changes so it can have at most four positive real roots. Now if we see P5(-x), it has only one sign change as
P5(-x) = -x5+2x4+x3+x2+x+2 = 0
Hence, it can have only one negative real root.
Please mark it as brainliest answer.
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