Explain the location of roots with explain and with all conditions and cases to that applied
Explain the location of roots with explain and with all conditions and cases to that applied
quadratic equation ax^2 + bx + c = 0
Generally we denote discriminant of the quadratic equation by ‘∆ ‘ or ‘D’.
Therefore,
Discriminant ∆ = b^2 - 4ac
Depending on the discriminant we shall discuss the following cases about the nature of roots α and β of the quadratic equation ax^2 + bx + c = 0.
When a, b and c are real numbers, a ≠ 0
Case I: b^2 - 4ac > 0
When a, b and c are real numbers, a ≠ 0 and discriminant is positive (i.e., b^2 - 4ac > 0), then the roots α and β of the quadratic equation ax^2 + bx + c = 0 are real and unequal.
Case II: b^2 - 4ac = 0
When a, b and c are real numbers, a ≠ 0 and discriminant is zero (i.e., b^2 - 4ac = 0), then the roots α and β of the quadratic equation ax^2 + bx + c = 0 are real and equal.
Case III: b^2 - 4ac < 0
When a, b and c are real numbers, a ≠ 0 and discriminant is negative (i.e., b^2 - 4ac < 0), then the roots α and β of the quadratic equation ax^2 + bx + c = 0 are unequal and imaginary. Here the roots α and β are a pair of the complex conjugates.
Case IV: b^2 - 4ac > 0 and perfect square
When a, b and c are real numbers, a ≠ 0 and discriminant is positive and perfect square, then the roots α and β of the quadratic equation ax^2 + bx + c = 0 are real, rational unequal.
Case V: b^2 - 4ac > 0 and not perfect square
When a, b and c are real numbers, a ≠ 0 and discriminant is positive but not a perfect square then the roots of the quadratic equation ax^2 + bx + c = 0 are real, irrational and unequal.
Here the roots α and β form a pair of irrational conjugates.
Case VI: b^2 - 4ac is perfect square and a or b is irrational
When a, b and c are real numbers, a ≠ 0 and the discriminant is a perfect square but any one of a or b is irrational then the roots of the quadratic equation ax^2 + bx + c = 0 are irrational.
Notes: (i) From Case I and Case II we conclude that the roots of the quadratic equation ax^2 + bx + c = 0 are real when b^2 - 4ac ≥ 0 or b^2 - 4ac ≮ 0.
(ii) From Case I, Case IV and Case V we conclude that the quadratic equation with real coefficient cannot have one real and one imaginary roots; either both the roots are real when
Various types of Solved examples on nature of the roots of a quadratic equation:
1. Find the nature of the roots of the equation 3x^2 - 10x + 3 = 0 without actually solving them.
Solution:
Here the coefficients are rational.
The discriminant D of the given equation is
D = b^2 - 4ac
= (-10)^2 - 4 ∙ 3 ∙ 3
= 100 - 36
= 64 > 0.
Clearly, the discriminant of the given quadratic equation is positive and a perfect square.
Therefore, the roots of the given quadratic equation are real, rational and unequal.
2. Discuss the nature of the roots of the quadratic equation 2x^2 - 8x + 3 = 0.
Solution:
Here the coefficients are rational.
The discriminant D of the given equation is
D = b^ - 4ac
= (-8)^2 - 4 ∙ 2 ∙ 3
= 64 - 24
= 40 > 0.
Clearly, the discriminant of the given quadratic equation is positive but not a perfect square.
Therefore, the roots of the given quadratic equation are real, irrational and unequal.
3. Find the nature of the roots of the equation x^2 - 18x + 81 = 0 without actually solving them.
Solution:
Here the coefficients are rational.
The discriminant D of the given equation is
D = b^2 - 4ac
= (-18)^2 - 4 ∙ 1 ∙ 81
= 324 - 324
= 0.
Clearly, the discriminant of the given quadratic equation is zero and coefficient of x^2 and x are rational.
Therefore, the roots of the given quadratic equation are real, rational and equal.
4. Discuss the nature of the roots of the quadratic equation x^2 + x + 1 = 0.
Solution:
Here the coefficients are rational.
The discriminant D of the given equation is
D = b^2 - 4ac
= 1^2 - 4 ∙ 1 ∙ 1
= 1 - 4
= -3 > 0.
Clearly, the discriminant of the given quadratic equation is negative.
Therefore, the roots of the given quadratic equation are imaginary and unequal.
quadratic equation ax^2 + bx + c = 0
Generally we denote discriminant of the quadratic equation by ‘∆ ‘ or ‘D’.
Therefore,
Discriminant ∆ = b^2 - 4ac
Depending on the discriminant we shall discuss the following cases about the nature of roots α and β of the quadratic equation ax^2 + bx + c = 0.
When a, b and c are real numbers, a ≠ 0
Case I: b^2 - 4ac > 0
When a, b and c are real numbers, a ≠ 0 and discriminant is positive (i.e., b^2 - 4ac > 0), then the roots α and β of the quadratic equation ax^2 + bx + c = 0 are real and unequal.
Case II: b^2 - 4ac = 0
When a, b and c are real numbers, a ≠ 0 and discriminant is zero (i.e., b^2 - 4ac = 0), then the roots α and β of the quadratic equation ax^2 + bx + c = 0 are real and equal.
Case III: b^2 - 4ac < 0
When a, b and c are real numbers, a ≠ 0 and discriminant is negative (i.e., b^2 - 4ac < 0), then the roots α and β of the quadratic equation ax^2 + bx + c = 0 are unequal and imaginary. Here the roots α and β are a pair of the complex conjugates.
Case IV: b^2 - 4ac > 0 and perfect square
When a, b and c are real numbers, a ≠ 0 and discriminant is positive and perfect square, then the roots α and β of the quadratic equation ax^2 + bx + c = 0 are real, rational unequal.
Case V: b^2 - 4ac > 0 and not perfect square
When a, b and c are real numbers, a ≠ 0 and discriminant is positive but not a perfect square then the roots of the quadratic equation ax^2 + bx + c = 0 are real, irrational and unequal.
Here the roots α and β form a pair of irrational conjugates.
Case VI: b^2 - 4ac is perfect square and a or b is irrational
When a, b and c are real numbers, a ≠ 0 and the discriminant is a perfect square but any one of a or b is irrational then the roots of the quadratic equation ax^2 + bx + c = 0 are irrational.
Notes:(i) From Case I and Case II we conclude that the roots of the quadratic equation ax^2 + bx + c = 0 are real when b^2 - 4ac ≥ 0 or b^2 - 4ac ≮ 0.
(ii) From Case I, Case IV and Case V we conclude that the quadratic equation with real coefficient cannot have one real and one imaginary roots; either both the roots are real when
Various types of Solved examples on nature of the roots of a quadratic equation:
1. Find the nature of the roots of the equation 3x^2 - 10x + 3 = 0 without actually solving them.
Solution:
Here the coefficients are rational.
The discriminant D of the given equation is
D = b^2 - 4ac
= (-10)^2 - 4 ∙ 3 ∙ 3
= 100 - 36
= 64 > 0.
Clearly, the discriminant of the given quadratic equation is positive and a perfect square.
Therefore, the roots of the given quadratic equation are real, rational and unequal.
2. Discuss the nature of the roots of the quadratic equation 2x^2 - 8x + 3 = 0.
Solution:
Here the coefficients are rational.
The discriminant D of the given equation is
D = b^ - 4ac
= (-8)^2 - 4 ∙ 2 ∙ 3
= 64 - 24
= 40 > 0.
Clearly, the discriminant of the given quadratic equation is positive but not a perfect square.
Therefore, the roots of the given quadratic equation are real, irrational and unequal.
3. Find the nature of the roots of the equation x^2 - 18x + 81 = 0 without actually solving them.
Solution:
Here the coefficients are rational.
The discriminant D of the given equation is
D = b^2 - 4ac
= (-18)^2 - 4 ∙ 1 ∙ 81
= 324 - 324
= 0.
Clearly, the discriminant of the given quadratic equation is zero and coefficient of x^2 and x are rational.
Therefore, the roots of the given quadratic equation are real, rational and equal.
4. Discuss the nature of the roots of the quadratic equation x^2 + x + 1 = 0.
Solution:
Here the coefficients are rational.
The discriminant D of the given equation is
D = b^2 - 4ac
= 1^2 - 4 ∙ 1 ∙ 1
= 1 - 4
= -3 > 0.
Clearly, the discriminant of the given quadratic equation is negative.
Therefore, the roots of the given quadratic equation are imaginary and unequal.
