QuizSolver
  • Bank PO
  • CBSE
  • IIT JEE
 
 

Grasp the concept of Position of roots of Quadratic Equations

Posted on - 19-02-2017

JEE Math QE

IIT JEE

Interval in which the Roots Lie

In some problems we want the roots of the equation ax2 + bx + c = 0 to lie in a given interval. For this we impose conditions on a, b and c. Since a > 0, we can take .

f(x) = x2 + .

(i) If both the roots are positive i.e. they lie in (0, ¥), then the sum of the roots as well as the product of the roots must be positive.


&⇒
a + b = -and ab = with b2 - 4ac ≥ 0.

Similarly, if both the roots are negative i.e. they lie in (- ¥, 0) then the sum of the roots must be negative and the product of the roots must be positive.

i.e. a + b = -< 0 and ab = with b2 – 4ac ≥ 0.

Both the roots are of the same sign if a and c are of same sign. Now if b has the same sign that of a, both roots are negative or else both roots are positive.

If a and c are of opposite sign both roots are of opposite sign.

(ii) Both the roots are greater than a given number k if the following three conditions are satisfied D ≥ 0, -and f(k) > 0.

(iii) Both the roots will be less than a given number k if the following conditions are satisfied: D ≥ 0, -< k and f(k) > 0.

(iv) Both the roots will lie in the given interval (k1, k2) if the following conditions are satisfied: D ≥ 0 k1 < -and f(k1) > 0, f(k2) > 0.

­­­

(v) Exactly one of the roots lies in the given interval (k1, k2) if f(k1) . f(k2) < 0.

(vi) A given number k will lie between the roots if f(k) < 0.

In particular, the roots of the equation will be of opposite signs if 0 lies between the roots
&⇒
f(0) < 0
. .

Example 1

Let x2 - (m - 3)x + m = 0(m ∈ R) be a quadratic equation. Find the values of m for which the roots are.

Solution:

 
Quadratic Equations - Solved Objective Questions Part 2 for Conceptual Clarity
Quadratic Equations - Solved Objective Questions Part 1 for Conceptual Clarity
Solved Objective Question on Probability Set 2
Solved Objective Question on Probability Set 1
Solved Objective Question on Progression and Series Set 2
Solved Objective Question on Permutations and Combinations Set 3
Solved Objective Question on Permutations and Combinations Set 2
Solved Objective Question on Progression and Series Set 1
Solved Objective Question on Permutations and Combinations Set 1
Quadratic Equations - Solved Subjective Questions Part 4
Quadratic Equations - Solved Subjective Questions Part 2
Quadratic Equations - Solved Subjective Questions Part 3
Quadratic Equations - Solved Subjective Questions Part 1
Solved Subjective Questions on Circle Set 9
Solving Equations Reducible to Quadratic Equations
Theory of Polynomial Equations and Remainder Theorem
Solved Subjective Questions on Circle Set 8
Solving Quadratic Inequalities Using Wavy Curve Methods
Basics of Quadratic Inequality or Inequations
Division and Distribution of Objects - Permutation and Combination

Comments