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Lesson Posted on 24 May Exam Coaching/Engineering Entrance Coaching CBSE/Class 11/Science/Mathematics CBSE/Class 12/Science/Mathematics

Nature of Roots ( Quadratic Equation

Pulkit

I have rich experience of a decade of guiding and mentoring students across the country who are aspiring...

NATURE OF ROOTS In quadratic equation ax2 + bx + c = 0, the term b2 – 4ac is called discriminant (D or D) . (1) If a, b, c Î R and a ¹ 0, then : (i) If D > 0, then roots are... read more

NATURE OF ROOTS                                                                                                   

In quadratic equation ax2 + bx + c = 0, the term b2 – 4ac is called discriminant (D or D) .

(1)    If a, b, c Î R and a ¹ 0, then :

(i)   If D > 0, then roots are real and distinct                     (ii)   If D = 0, then roots are real and equal

(iii)  If D ³ 0, then roots are real .                                   (iv)  If D < 0, then roots are complex .

(2)    If a, b, c Î Q, a ¹ 0, then :

(i)   If D > 0 and D is a perfect square then roots are rational.

(ii) If D > 0 and D is not a perfect square then roots are irrational .

(3)    Conjugate roots: The irrational and complex roots of a quadratic equation always occur in pairs. Therefore

(i)   If one root be a + ib then other root will be a - ib.

(ii)   If one root be a + Öb then other root will be a - Öb.    

(4)    If D1 and D2 be the discriminants of two quadratic equations, then

(i)   If D1 + D2 ³ 0, then (a) At least one of D1 and D2 ³ 0.            (b) If D1 < 0 then D2 > 0

(ii) If D1 + D2 < 0, then   (a) At least one of D1 and D2 < 0.            (b) If D1 > 0 then D2 < 0.

 

Brain Demur…

The quadratic trinomial ax2 + bx + c = 0 will be a perfect square if D = 0 i.e. b2 – 4ac = 0

If a is a repeated root of the ax2 + bx + c = 0 then a is a root of equation f’(x) = 0 as well.

If a, b, c ÏR then the roots need not be conjugate.

If a, b, c are irrational then the roots need not be conjugate.

If a + b + c = 0 then one root is always unity and the other root is , c/a

 

ROOTS UNDER PARTICULAR CONDITIONS                                                               

For the quadratic equation ax2 + bx + c = 0.

(1)     b = 0 Þ roots are of equal magnitude but of opposite sign.

(2)     c = 0 Þ one root is zero, other is –b/a.

(3)     b = c = 0 Þ both roots are zero.

(4)     a = c Þ roots are reciprocal to each other.

(5)     a + b + c = 0 Þ one root is 1 and second root is c/a.

(6)     a = b = c = 0, then equation will become an identity and will be satisfied by every value of x.

(7)     a = 1 and b, c Î I and the root of equation ax2 + bx + c = 0 are rational numbers, then these roots must be integers.

 

 

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Asked on 20 May CBSE/Class 12/Science/Mathematics

Why we Differentiate between different functions?

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Lesson Posted on 14 May CBSE/Class 12/Science/Mathematics Exam Coaching/UPSC Exams Coaching/NDA (National Defence Acadamy and Naval Acadamy) Exam Coaching Tuition/Class VI-VIII Tuition/English

Preparing for Maths Exam for SAT, GRE, NDA-NA & VII to XII Standard of Any Board

Arun Vinayak Joshi

After completing my BE, I gathered 33 years of industry experience by working in a variety of small-scale...

First, prepare a Formulae Notebook to make Ready Reference for solving the problems. Prepare Index page, write chapter numbers and number all pages of Notebook. Write Chapterwise formulae and mention respective chapter-wise page numbers in the Index page. Start solving problems in a separate notebook... read more

First, prepare a Formulae Notebook to make Ready Reference for solving the problems.

Prepare Index page, write chapter numbers and number all pages of Notebook.

Write Chapterwise formulae and mention respective chapter-wise page numbers in the Index page.

Start solving problems in a separate notebook keeping Formulae Notebook handy. Write all the steps and calculations.

If your answers do not match with the answers given, first check your steps and calculations for errors, and correct them. If it does not match, then only refer to the Answers & Explanations or guide-book.

Give priority to Correctness. Once the practise exercises are over, Go ahead for question paper set or test series. Keep away Formulae Notebook and keep alarm clock to alert you about Exam Time of 3 hours or as the case may be. This way you can get the assessment of your speed.

Some necessary Tables of 2 to 30, Squares of 1 to 30, Cubes of 1 to 30, Rules of divisibility by numbers 2,3,4,5,9 & 11 may also be kept handy if forgotten.

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Asked on 02 May CBSE/Class 12/Science/Mathematics

There was a mistake while typing my phone number, and now the changed no is not being accepted stating... read more

There was a mistake while typing my phone number, and now the changed no is not being accepted  stating the no is already in use how to change as it is not accepting a fresh application

 

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Asked on 01 May CBSE/Class 12/Science

I am in class 12th in science stream i am confused i am not able to decide in which field i should move... read more

I am in class 12th in science stream i am confused i am not able to decide in which field i should move after 12th i am not interested in engineering please suggest me ways......

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Asked on 29 Apr CBSE/Class 12/Science/Mathematics

Hi I am looking for a good maths academy in Chennai, most preferably triplicane area. Can you suggest... read more

Hi I am looking for a good maths academy in  Chennai, most preferably triplicane area. Can you suggest me a good institution 

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Looking for Class XI-XII Tuition (PUC)

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Asked on 15 Apr CBSE/Class 12/Science/Mathematics

Please send me in reply how to draw trigonometry table 

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Answered on 08 Apr CBSE/Class 12/Science CBSE/Class 12/Science/Chemistry Organic Chemistry

What is one of the most appropriate Chemistry Study Books for Class XII ?

Somnath Chowdhury

Chemistry Tutor

KL Chugh is good book for beginers. Adter that JD lee for inorganic, R K Gupta for organic and Willey publication physical chemistry for JEE will be appropriate.
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Answered on 29/01/2017 CBSE/Class 12/Science/Mathematics Tuition/Class XI-XII Tuition (PUC)

Prashanth K

Technical Trainer

Row and column wise arrangement of data.
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