Quadratic equations for IIT-JEE MAINS /ADVANCED

BASIC RESULTS RELATED TO A QUADRATIC EQUATIONS

quadratic 1

The quantity D

quadratic 2

  • The quadratic equation has real and equal roots if and only if D = 0

 

  • The quadratic equation has real and distinct roots if and only if D > 0

 

  • The quadratic equation has complex roots with non-zero imaginary parts if and only if D < 0 . If p + iq (p and q being real) is a root of the quadratic equation where i = , then p – iq is also a root of the quadratic equation.

 

CONCEPT OF IDENTITY

If the quadratic equation is satisfied by more than two distinct numbers (real or complex), then it becomes an identity i.e. a = b = c = 0. For example

quadratic identity-1

is satisfied by three value of x which are a, b and c. Hence this is an identity in x.

quadratic identity

 

HOW TO SOlVE RATIONAL INEQUALITIES

In order to solve inequalities of the form

quadratic.1

we use the following method:

If x1 and x2 (x1 < x2) are two consecutive distinct roots of a polynomial equation, then within this
interval the polynomial itself takes on values having the same sign. Now find all the roots of the polynomial equations P(x) = 0 and Q(x) = 0. Ignore the common roots and write

quadratic equation 2
Where a1, a2, . . . . . an, b1, b2, . . . . . , bm are distinct real numbers. Then f(x) = 0 for x = a1,
a2, . . . . . , an and f(x) is not defined for x = b1, b2, . . . . . , bm. Apart from these (m + n) real
numbers f(x) is either positive or negative. Now arrange a1, a2, . . . . . , an, b1, b2, . . . . . , bm in an
increasing order say c1, c2, c3, c4, c5, . . . . . , cm+n. Plot them on the real line. Draw a curve starting
from right of cm+n along the real line which alternately changes its position at these points. This curve
is known as the wavy curve.
quadratic equation 3
The intervals in which the curve is above the real line will be the intervals for which f(x) is positive
and the intervals in which the curve is below the real line will be the intervals in which f(x) is
negative.

Other way of understanding the same concept

  • Write the inequality in the correct form. One side must be zero and the other must be in product form
  • Find the key or critical values. To find the key/critical values, set the numerator and denominator of the fraction equal to zero and change them in product of linear forms
  • Make a sign analysis chart. To make a sign analysis chart, use the key/critical values found in Step 2 to and mention them on line in increasing form.
  • Perform the sign analysis. and make wave that give the sign of product
  • What is needed acording to that write final answers
  • Problems related to rational inequalities with video solutions

 

CONDITIONS OF COMMON ROOTS FOR TWO QUADRATIC EQUATIONS

For two quadratic equations when one root is common / When both roots are common are given

quadratic equation - common roots-1

HOW TO FIND  RANGE OF A RATIONAL EXPRESSION IN X, WHERE X IS REAL

   Put the given rational expression equal to y and form the quadratic equation in x.

Find the discriminant D of the quadratic equation obtained in step 1.

  Since x is real, therefore, put D   GREATER THAN OR EQUAL TO 0. We get an inequation in y.

      Solve the above inequation for y. The range of y so obtained determines the range attained by the given rational expression

RANGE WITH HELP OF QUADRATIXC EQUATION

 

QUADRATIC EXPRESSION-LOCATION OF ROOTS

The expression ax2 + bx + c is said to be a real quadratic expression in x where a, b, c are real and
a 0. Let f(x) = ax2 + bx + c, where a, b, c R (a 0). f(x) can be re-written as

quadratic expression 1
is the discriminant of the quadratic expression.

Therefore y = f(x) represents a parabola whose axis is parallel to the y-axis, with vertex at
A .

quadratic expression 2
Note that if a > 0, the parabola will be concave upwards and if a < 0 the parabola will be concave
downwards and it depends on the sign of D that the parabola cuts the x-axis at two points
(D > 0), touches the x-axis (D = 0) or never intersects with the x-axis (D< 0).
This gives rise to the following cases:
quadratic expression 3
In this case the parabola always remains concave upwards and above the x-axis

quadratic expression 4

quadratic expression 5
In this case the parabola cuts the x-axis at two points and and remains concave
upwards.

(iv)

quadratic expression 6

a < 0 and D< 0
f(x) < 0xR.
In this case the parabola remains concave downwards and always below the x-axis.

quadratic expression 7
In this case the parabola touches the x-axis and remains concave downwards.

quadratic expression 9

RELATION BETWEEN THE ROOTS OF A POLYNOMIAL EQUATION OF DEGREE N AND coefficient

relation between coefficient nd roots 1

relation between coefficient nd roots 2

IMPORTANT POINTS

A polynomial equation of degree n has n roots (real or imaginary).

If all the coefficients are real then the imaginary roots occur in pairs i.e., number of imaginary roots is always even.

If the degree of a polynomial equation is odd then the number of real roots will also be odd. It follows that at least one of the roots will be real.

Polynomial in x of degree n can be factorized into a product of linear/quadratic form.

I

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