Intervals In Which The Roots Of A Quadratic Equation Lie

IIT JEE Quadratic Equation study notes are designed to help the students to understand several important concepts included in IIT JEE quadratic Equation syllabus like finding the range of a Quadratic Equation, determining the Intervals in which the roots of a quadratic Equation will lie, the maximum and minimum value of a quadratic equation and more. These IIT JEE Quadratic Equation study notes have a collective extract from several books and contain all the important tips helpful for solving typical mathematical problems asked in IIT JEE Exams accurately. The students can easily understand and practice important quadratic equation IIT JEE concepts by referring to these study notes.

  1. Introduction to Quadratic Equations
  2. Quadratic Inequalities
  3. Graphing Quadratic Equations
  4. Range of a Quadratic Equation: Maximum and Minimum Values
  5. Theory of Equations

Consider a Quadratic expression \(f(x) = ax^{2} + bx + c\), where a ≠ 0 and a, b, and c are real. The quadratic Expression can be further rewritten as f (x) = \(\mathbf{x^{2} \;+\;\frac{b}{a}\;x\;=\;\frac{c}{a}}\).

Case 1: Both the roots of a quadratic Expression (α, β) are greater than any given number ‘m’ if,

1. \(b^{2} – 4ac = (D) ≥ 0\),

2. \(\mathbf{\frac{-b}{2a}}\) > m,

3. f (m) > 0.

Quadratic Equation IIT JEE

Case 2: Both the roots of a quadratic Expression (α, β) are less than any given number ‘m’ if,

1. \(b^{2} – 4ac = (D) ≥ 0\),

2. \(\mathbf{\frac{-b}{2a}}\) < m,

3. f (m) > 0.

Quadratic Equation IIT JEE

Case 3: Both the roots of a quadratic Expression (α, β) will lie in the given interval (\(m_{1}\), \(m_{2}\)) if,

1. \(b^{2} – 4ac = (D) ≥ 0\),

2. \(m_{1}\) < \(\mathbf{\frac{-b}{2a}}\) >\(m_{2}\),

3. f (\(m_{1}\)) > 0,

4. f (\(m_{2}\)) > 0.

Quadratic Equation IIT JEE

Case 4: If exactly one root of a quadratic equation (α, β) will lie in the given interval (\(m_{1}\), \(m_{2}\)) if,

f (\(m_{1}\)) . f (\(m_{2}\)) < 0

Quadratic Equation IIT JEE

Case 5: The given number ‘m’ will lie between the roots of a quadratic Equation α and β if,

f (m) < 0

Quadratic Equation IIT JEE

Case 6: The roots of a quadratic Equation α and β will have have opposite sign if,

f (0) < 0.

Case 7: Both the roots of a quadratic Expression α and β are positive if,

1. \(b^{2} – 4ac = (D) ≥ 0\),

2. α + β = \(\mathbf{\frac{-b}{a}}\) > 0,

3. αβ = \(\mathbf{\frac{c}{a}}\) > 0.

Quadratic Equation IIT JEE

Case 8: Both the roots of a quadratic Expression α and β are negative if,

1. \(b^{2} – 4ac = (D) ≥ 0\),

2. α + β = \(\mathbf{\frac{-b}{a}}\) < 0,

3. αβ = \(\mathbf{\frac{c}{a}}\) < 0.

Quadratic Equation IIT JEE

Quadratic Equation IIT JEE Problems:

Example 1: Find the range of k for which 6 lies between the roots of the quadratic equation \(x^{2} + 2 (k – 3) x + 9 = 0\).

Solution:

6 will lie between the roots of the quadratic expression f(x) = \(x^{2} + 2 (k – 3) x + 9\) if,

f (6) < 0

i.e. 36 + 2 (k – 3) . 6 + 9 < 0,

= 36 + 12k – 36 + 9 < 0,

= k < \(\mathbf{-\;\frac{3}{4}}\)

Therefore, the range of k for which 6 lies between the roots of the given quadratic equation is:

\(\mathbf{k\;\in \;\left ( -\;\infty ,\; -\;\frac{3}{4} \right )}\)

Example 2: Find the values of k for which exactly one root of the quadratic equation \(4 x^{2} – 4 (k – 2) x + k – 2 = 0\) lies in \(\mathbf{\left ( 0 ,\;\frac{1}{2} \right )}\).

Solution:

Exactly one root of the quadratic expression \(4 x^{2} – 4 (k – 2) x + k – 2 = 0\) will lie in the given interval if,

f (\(m_{1}\)) . f (\(m_{2}\)) < 0

i.e. \(\mathbf{f\;\left ( 0 \right )\;.\;f\;\left ( \frac{1}{2} \right )\;<\; 0}\)

Or, (k – 2) [1 – 2 (k – 2)+ k – 2] < 0,

Or, (k -2) (3 – k) < 0,

Or, (k – 2) (k – 3) > 0.

Therefore, the values of k for which exactly one root of the given quadratic equation will lie in the interval \(\mathbf{\left ( 0 ,\;\frac{1}{2} \right )}\):

\(\mathbf{k\;\in \;\left ( -\;\infty ,\;2 \right )\;\cup \;\left ( 3,\; \infty \right )}\)

Example 3. Find the values of k for which the roots of the quadratic equation \(x^{2} – (k – 3) x + k = 0\) are greater than 2.

Solution:

The roots of the given quadratic expression f(x) = \(x^{2} – (k – 3) x + k\) are greater than 2 if,

Condition 1: \(\mathbf{-\;\frac{b}{2\;a}\;>\; 2}\)

i.e. \(\mathbf{\frac{+\left ( k\;-\;3 \right )}{2}\;>\; 2}\)

Therefore, k > 7 . . . . . . . . . . . . . (1)

Condition 2: f (2) > 0

i.e. \(4 – (k – 3)^{2} + k > 0\),

Or, 4 – 2k + 6 + k > 0.

Therefore, k < 10 . . . . . . . . . . . . . . (2)

Condition 3: \(b{2} – 4ac ≥ 0\)

i.e. \((k – 3)^{2} – 4k ≥ 0,\)

Or, \(k^{2} + 10k + 9 ≥ 0\),

Or, (k – 9) (k – 1) ≥ 0.

Therefore, \(\mathbf{k\;\in \;\left (-\;\infty ,\; 1 \right ]\;\cup \;\left [ 9, \;\infty \right )}\) . . . . . . . . . (3)

From Equation (1), (2) and, (3) the range of ‘k’ for which the roots of the given quadratic expression are greater than 2:

\(\mathbf{k\;\in \;[9, \;10)}\)

Example 4: Find the values of ‘m’ for which the roots of the given quadratic expression \(f (x) = 9x^{2} + (m – 4) x + \frac{m}{4}\) satisfies the following given conditions:

1: Both the roots of quadratic expression f (x) are real and distinct.

2: Quadratic Expression f (x) has equal roots.

3: Roots of quadratic expression are not real.

4: Roots of quadratic Equation have opposite sign.

5: Roots are equal in magnitude but opposite sign.

6: Both the roots of the given quadratic expression f (x) are positive.

7: Both of the roots are negative.

Solution:

From the given quadratic Expression\(f (x) = 9x^{2} + (m – 4) x + \frac{m}{4}\),

a = 9, b = (m – 4) and c = \(\mathbf{\frac{m}{4}}\)

Therefore, D = \(b^{2} – 4ac\) = \((m – 4)^{2} – 4 (9) \left ( \frac{m}{4} \right )\)

= \(m^{2} + 16 – 8m – 9m\)

= \(m^{2} – 17m + 16\)

Therefore, \(D = m^{2} -17m + 16 = (m – 1) (m – 16)\)

1. Both the roots of the Quadratic Equations are real and Distinct if,

D = (\(b^{2} – 4ac\)) > 0

i.e (m – 16) (m – 1) > 0

Note: If \(ax^{2} + bx + c > 0\), D > 0 and a > 0, then \(\mathbf{x\;\in \;\left ( -\;\infty ,\; \alpha \right )\;\cup \;\left ( \beta ,\;\infty \right )}\).

Therefore, both the roots of given quadratic equation are real and distinct if:

\(\mathbf{m\;\in \;\left ( -\;\infty ,\; 1 \right )\;\cup \;\left ( 16 ,\;\infty \right )}\).

2. Quadratic Expression f (x) has equal roots if, D = 0

i.e. (m – 16) (m – 1) = 0

Therefore, for m = 16 or m = 1, both the roots of given quadratic equation are equal.

3. Roots of quadratic expression are imaginary if, D < 0

i.e. (m -16) (m – 1) < 0

Note: If \(ax^{2} + bx + c > 0\), D < 0 and a > 0, then \(\mathbf{x\;\in \;R}\)

Therefore, the roots of quadratic equation are imaginary if, \(\mathbf{m\;\in \;(1,\;16)}\).

4. Roots of quadratic expression will have opposite sign if, f (0) < 0

Since f (0) = m

Therefore, the roots of quadratic equation will have opposite sign if \(\mathbf{m\;\in \;(-\;\infty,\;0)}\).

5. The roots of quadratic expression are equal in magnitude but opposite sign if,

Condition 1: Sum of roots = 0

i.e. \(\mathbf{ \frac{m\;-\;4}{9}\;=\;0}\)

Therefore, m = 4 . . . . . . . . . . . . . . . (1)

Condition 2: D = (\(b^{2} – 4ac\)) ≥ 0

i.e. (m -1) (m – 16) ≥ 0

Therefore, \(\mathbf{x\;\in \;\left ( -\;\infty ,\; 1 \right ]\;\cup \;\left [ 16 ,\;\infty \right )}\) . . . . . . . . . . . . . . . . (2)

From Equation (1) and (2), we conclude that no such value of ‘m’ exist in f (x) when the roots have opposite sign with equal magnitude.

6. Both the roots of given quadratic expression are positive if,

Condition 1: Product of the roots > 0

i.e. \(\mathbf{ \frac{m}{4\;\times \;9}\;> \;0}\)

Therefore, m > 0 . . . . . . . . . . (3)

Condition 2: Sum of the roots > 0

\(\mathbf{ \frac{-\;(m\;-4)}{9}\;> \;0}\)

Therefore, m < 4 . . . . . . . . . (4)

Condition 3: D = (\(b^{2} – 4ac\)) ≥ 0

From Equation (2),

\(\mathbf{x\;\in \;\left ( -\;\infty ,\; 1 \right ]\;\cup \;\left [ 16 ,\;\infty \right )}\) . . . . . . . . (5)

Therefore, from Equation (3), (4), and (5) both the roots of f (x) are positive if,

\(\mathbf{m\;\in \;(0,\;1]}\)

7. Both the roots are negative if,

Condition 1: The Product of roots > 0

i.e. \(\mathbf{ \frac{m}{4\;\times \;9}\;> \;0}\)

i.e. m > 0 . . . . . . . . . . (6)

Condition 2: The Sum of roots < 0

\(\mathbf{ \frac{-\;(m\;-4)}{9}\;< \;0}\)

i.e. m > 4 . . . . . . . . . (7)

Condition 3: D = (\(b^{2} – 4ac\)) ≥ 0

Using Equation (2),

\(\mathbf{x\;\in \;\left ( -\;\infty ,\; 1 \right ]\;\cup \;\left [ 16 ,\;\infty \right )}\) . . . . . . . . (8)

Therefore, from Equations (6), (7), and (8) both the roots of quadratic expression f (x) are negative if,

\(\mathbf{m\;\in \;[16,\;\infty ]}\)


Practise This Question

Besides the risk of pollution, fossil fuels also pose a risk of ___