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 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

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

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

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

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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