We will discuss here about the different cases of discriminant to understand the nature of the roots of a quadratic equation.
We know that α and β are the roots of the general form of the quadratic equation ax\(^{2}\) + bx + c = 0 (a ≠ 0) .................... (i) then we get
α = \(\frac{- b - \sqrt{b^{2} - 4ac}}{2a}\) and β = \(\frac{- b + \sqrt{b^{2} - 4ac}}{2a}\)
Here a, b and c are real and rational.
Then, the nature of the roots α and β of equation ax\(^{2}\) + bx + c = 0 depends on the quantity or expression i.e., (b\(^{2}\) - 4ac) under the square root sign.
Thus the expression (b\(^{2}\) - 4ac) is called the discriminant of the 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 b\(^{2}\) - 4ac > 0 or both the roots are imaginary when b\(^{2}\) - 4ac < 0.
(iii) From Case IV and Case V we conclude that the quadratic equation with rational coefficient cannot have only one rational and only one irrational roots; either both the roots are rational when b\(^{2}\) - 4ac is a perfect square or both the roots are irrational b\(^{2}\) - 4ac is not a perfect square.
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\(^{2}\) - 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.
Or,
The roots of the given equation are a pair of complex conjugates.
11 and 12 Grade Math
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