BASIC RESULTS RELATED TO A QUADRATIC EQUATIONS
The quantity D
 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 nonzero 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
is satisfied by three value of x which are a, b and c. Hence this is an identity in x.
HOW TO SOlVE RATIONAL INEQUALITIES
In order to solve inequalities of the form
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
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
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
QUADRATIC EXPRESSIONLOCATION 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 rewritten as
Therefore y = f(x) represents a parabola whose axis is parallel to the yaxis, with vertex at
A .
(iv)
a < 0 and D< 0
f(x) < 0xR.
In this case the parabola remains concave downwards and always below the xaxis.
RELATION BETWEEN THE ROOTS OF A POLYNOMIAL EQUATION OF DEGREE N AND coefficient
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.
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Very good post for quadratic equations