Please tell me about ....DISCRIMINANT AND NATURE OF ROOTS PPZZ FAST..
Discriminant ∆ = b2 - 4ac
Depending on the discriminant we shall discuss the following cases about the nature of roots α and β of the quadratic equation ax2 + bx + c = 0.
When a, b and c are real numbers, a ≠ 0
Case I: b2 - 4ac > 0
When a, b and c are real numbers, a ≠ 0 and discriminant is positive (i.e., b2 - 4ac > 0), then the roots α and β of the quadratic equation ax2 + bx + c = 0 are real and unequal.
Case II: b2 - 4ac = 0
When a, b and c are real numbers, a ≠ 0 and discriminant is zero (i.e., b2 - 4ac = 0), then the roots α and β of the quadratic equation ax2 + bx + c = 0 are real and equal.
Case III: b2 - 4ac < 0
When a, b and c are real numbers, a ≠ 0 and discriminant is negative (i.e., b2 - 4ac < 0), then the roots α and β of the quadratic equation ax2 + bx + c = 0 are unequal and imaginary. Here the roots α and β are a pair of the complex conjugates.
Case IV: b2 - 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 ax2 + bx + c = 0 are real, rational unequal.
Case V: b2 - 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 ax2 + bx + c = 0 are real, irrational and unequal.
Here the roots α and β form a pair of irrational conjugates.
Case VI: b2 - 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 ax2 + bx + c = 0 are irrational.
Discriminant ∆ = b2 - 4ac
Depending on the discriminant we shall discuss the following cases about the nature of roots α and β of the quadratic equation ax2 + bx + c = 0.
When a, b and c are real numbers, a ≠ 0
Case I: b2 - 4ac > 0
When a, b and c are real numbers, a ≠ 0 and discriminant is positive (i.e., b2 - 4ac > 0), then the roots α and β of the quadratic equation ax2 + bx + c = 0 are real and unequal.
Case II: b2 - 4ac = 0
When a, b and c are real numbers, a ≠ 0 and discriminant is zero (i.e., b2 - 4ac = 0), then the roots α and β of the quadratic equation ax2 + bx + c = 0 are real and equal.
Case III: b2 - 4ac < 0
When a, b and c are real numbers, a ≠ 0 and discriminant is negative (i.e., b2 - 4ac < 0), then the roots α and β of the quadratic equation ax2 + bx + c = 0 are unequal and imaginary. Here the roots α and β are a pair of the complex conjugates.
Case IV: b2 - 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 ax2 + bx + c = 0 are real, rational unequal.
Case V: b2 - 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 ax2 + bx + c = 0 are real, irrational and unequal.
Here the roots α and β form a pair of irrational conjugates.
Case VI: b2 - 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 ax2 + bx + c = 0 are irrational.
