About Discriminant Formulas
To find the Discriminant of a polynomial equation, apply the discriminant formulae. The discriminant of a quadratic equation is particularly useful for determining the number and character of the roots. Discriminant of a polynomial is a function made out of the polynomial's coefficients. Get the list of Maths Formulas.
What Is Discriminant Formulas?
The discriminant formulae provide an understanding of the roots' nature. The quadratic formula yields the discriminant of a quadratic equation. The discriminant is represented by the letters D or Δ. For a quadratic equation and a cubic equation, the discriminant formulas are:
Discriminant Formula for a Quadratic Equation:
D = b2 - 4ac is the discriminant formula for the quadratic equation ax2 + bx + c = 0. Because the degree of a quadratic equation is 2, we know that it can only have two roots. The quadratic formula (ax2 + bx + c = 0) is used to discover the roots of a quadratic equation. x = [-b (b2 - 4ac)] / [2a] is a quadratic formula that can be used to get the roots. The discriminant D is b2 - 4ac, and it is inside the square root. As a result, the quadratic formula is x = [-b ± √D] / [2a]. D can be > 0, = 0, < 0 in this case. Let's figure out what the roots are in each of these scenarios.
If D > 0, the quadratic formula is x = [-b ± √(positive number)] / [2a], and the quadratic equation has two unique real roots in this case.
When D = 0, the quadratic formula becomes x = [-b] / [2a], and the quadratic equation has just one real root in this case.
If D is less than zero, the quadratic formula is x = [-b ± √(negative number)] / [2a]. Because the square root of a negative number produces an imaginary number, the quadratic equation has two distinct complex roots in this situation. For instance, √(-4) = 2i).
Discriminant Formula of a Cubic Equation
D = b2c2 − 4ac3 − 4b3d − 27a2d2 + 18abcd is the discriminant formula for a cubic equation ax3 + bx2 + cx + d = 0. Because the degree of a cubic equation is 3, we know that it can only have three roots. Here,
All the three roots are real and distinct if D > 0.
If D = 0, then all three roots are real, as long as at least two of them are equal.
If D is less than zero, two of its roots are complex numbers and the third is real.
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