Quadratic Equations

Question 1

Descartes Rule of Signs : The maximum number of positive and negative real roots are—

Answer 1

Maximum positive : number of changes of signs
Maximum negative: number of changes of signs taking (x) = (-x)

Question 2

1-If D is a perfect square::
2-If D is not a perfect square::

Answer 2

1- Roots are rational
2- Roots are irrational

Question 3

If a+b+c =0

Answer 3

Then 1 is one of the roots

Question 4

Imaginary roots are always in

Answer 4

Multiple of 2

Question 5

If a and c have opposite signs

Answer 5

Then it will have real roots

Question 6

If a and c have same sign and if b is 0

Answer 6

Then has imaginary roots

Question 7

Quadratic equation in factor form

Answer 7

a(x-alpha) (x- beta) =0

Question 8

Taking reciprocal of roots

Answer 8

New equation have a and c interchange

Question 9

One common root (both formulas)

Answer 9

1. alpha = -(b1c2 - b2c1) / (a1c2 -a2c1)
2. (a1c2 - a2c1)^2 = (a1b2 - a2b1) (b1c2 - b2c1)

Question 10

Both roots common

Answer 10

a1 / a2 = b1 / b2 = c1 / c2

Question 11

Write ans in inequality in interval form

Answer 11

Open interval : (a,b) = a < x < b
Close interval : [a,b] = a <_ x <_ b

Question 12

How to take < ,> in wavy curve method?

Answer 12

Equation > 0 : take +
Equation < 0 : take -

Question 13

What is the value of graph at vertex ?

Answer 13

At x axis : -b /2a
At y axis : -D /4a

Question 14

If a>0 and a<0 what is the minimum and maximum value of the expression

Answer 14

If a > 0 : maximum value: infinity
Minimum value: -D /4a
If a <0 : maximum value : -D /4a
Minimum value : infinity

NOTE : At any point value of the graph is always respective y axis

Question 15

Conditions for both +ve as well as -ve graph

Answer 15

1. a>0 , D>0
2. a<0 , D>0

Question 16

Conditions for only +ve and only -ve graph

Answer 16

Only +ve : a>0 , D<0
Only -ve : a<0 , D<0

Question 17

What are the conditions for equation when both roots are less than k ?

Answer 17

1. D>_0
2. -b /2a < k
3. a f(k) >0

Question 18

What are the conditions for the equation when both roots are greater than k ?

Answer 18

1. D>_0
2. -b /2a >k
3. a f(k) >0

Question 19

What is the condition for equation when k lies between the roots ?

Answer 19

a f(k) <0