How quadratic equation depends on discriminant?
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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 ax22 + bx + c = 0 (a ≠ 0) .................... (i) then we get
α = −b−b2−4ac√2a−b−b2−4ac2a and β = −b+b2−4ac√2a−b+b2−4ac2a
Here a, b and c are real and rational.
Then, the nature of the roots α and β of equation ax22 + bx + c = 0 depends on the quantity or expression i.e., (b22 - 4ac) under the square root sign.
Thus the expression (b22 - 4ac) is called the discriminant of the quadratic equation ax22 + bx + c = 0.
Generally we denote discriminant of the quadratic equation by ‘∆ ‘ or ‘D’.
Therefore,
Discriminant ∆ = b22 - 4ac
Depending on the discriminant we shall discuss the following cases about the nature of roots α and β of the quadratic equation ax22 + bx + c = 0.
When a, b and c are real numbers, a ≠ 0
Case I: b22 - 4ac > 0
When a, b and c are real numbers, a ≠ 0 and discriminant is positive (i.e., b22 - 4ac > 0), then the roots α and β of the quadratic equation ax22 + bx + c = 0 are real and unequal.
Case II: b22 - 4ac = 0
When a, b and c are real numbers, a ≠ 0 and discriminant is zero (i.e., b22 - 4ac = 0), then the roots α and β of the quadratic equation ax22 + bx + c = 0 are real and equal.
Case III: b22 - 4ac < 0
When a, b and c are real numbers, a ≠ 0 and discriminant is negative (i.e., b22 - 4ac < 0), then the roots α and β of the quadratic equation ax22 + bx + c = 0 are unequal and imaginary. Here the roots α and β are a pair of the complex conjugates.
Case IV: b22 - 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 ax22 + bx + c = 0 are real, rational unequal.
Case V: b22 - 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 ax22 + bx + c = 0 are real, irrational and unequal.
Here the roots α and β form a pair of irrational conjugates.
Case VI: b22 - 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 ax22 + bx + c = 0 are irrational.
Notes:
(i) From Case I and Case II we conclude that the roots of the quadratic equation ax22 + bx + c = 0 are real when b22 - 4ac ≥ 0 or b22 - 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 b22 - 4ac > 0 or both the roots are imaginary when b22 - 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 b22 - 4ac is a perfect square or both the roots are irrational b22 - 4ac is not a perfect square.
We know that α and β are the roots of the general form of the quadratic equation ax22 + bx + c = 0 (a ≠ 0) .................... (i) then we get
α = −b−b2−4ac√2a−b−b2−4ac2a and β = −b+b2−4ac√2a−b+b2−4ac2a
Here a, b and c are real and rational.
Then, the nature of the roots α and β of equation ax22 + bx + c = 0 depends on the quantity or expression i.e., (b22 - 4ac) under the square root sign.
Thus the expression (b22 - 4ac) is called the discriminant of the quadratic equation ax22 + bx + c = 0.
Generally we denote discriminant of the quadratic equation by ‘∆ ‘ or ‘D’.
Therefore,
Discriminant ∆ = b22 - 4ac
Depending on the discriminant we shall discuss the following cases about the nature of roots α and β of the quadratic equation ax22 + bx + c = 0.
When a, b and c are real numbers, a ≠ 0
Case I: b22 - 4ac > 0
When a, b and c are real numbers, a ≠ 0 and discriminant is positive (i.e., b22 - 4ac > 0), then the roots α and β of the quadratic equation ax22 + bx + c = 0 are real and unequal.
Case II: b22 - 4ac = 0
When a, b and c are real numbers, a ≠ 0 and discriminant is zero (i.e., b22 - 4ac = 0), then the roots α and β of the quadratic equation ax22 + bx + c = 0 are real and equal.
Case III: b22 - 4ac < 0
When a, b and c are real numbers, a ≠ 0 and discriminant is negative (i.e., b22 - 4ac < 0), then the roots α and β of the quadratic equation ax22 + bx + c = 0 are unequal and imaginary. Here the roots α and β are a pair of the complex conjugates.
Case IV: b22 - 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 ax22 + bx + c = 0 are real, rational unequal.
Case V: b22 - 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 ax22 + bx + c = 0 are real, irrational and unequal.
Here the roots α and β form a pair of irrational conjugates.
Case VI: b22 - 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 ax22 + bx + c = 0 are irrational.
Notes:
(i) From Case I and Case II we conclude that the roots of the quadratic equation ax22 + bx + c = 0 are real when b22 - 4ac ≥ 0 or b22 - 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 b22 - 4ac > 0 or both the roots are imaginary when b22 - 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 b22 - 4ac is a perfect square or both the roots are irrational b22 - 4ac is not a perfect square.
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