What is Discriminant? | Mathematics for Grade 12 PDF Download

Introduction

• Discriminant of a polynomial in math is a function of the coefficients of the polynomial. It is helpful in determining what type of solutions a polynomial equation has without actually finding them. i.e., it discriminates the solutions of the equation (as equal and unequal; real and nonreal) and hence the name "discriminant". It is usually denoted by Δ or D. The value of the discriminant can be any real number (i.e., either positive, negative, or 0).

Discriminant Formula

• The discriminant (Δ or D) of any polynomial is in terms of its coefficients. Here are the discriminant formulas for a cubic equation and quadratic equation.
  • The discriminant formulas of a quadratic equation.
    ax² + bx + c = 0 is
    Δ (or) D = b² - 4ac
  • The discriminant formulas of a cubic equation.
    ax² + b³x + cx² + d = 0 is
    Δ (or) D = b²c² - 4ac³ - 4b³d - 27a²d² + 18abcd

How to Find Discriminant?

Discriminant of a Quadratic Equation

• The discriminant of a quadratic equation ax² + bx + c = 0 is in terms of its coefficients a, b, and c. i.e.,
  • Δ OR D = b² − 4ac
• Do you recall using b² − 4ac earlier? Yes, it is a part of the quadratic formula:Here, the expression that is inside the square root of the quadratic formula is called the discriminant of the quadratic equation. The quadratic formula in terms of the discriminant is: x =

Example: Find the discriminant of the quadratic equation 2x² - 3x + 8 = 0.

Comparing the equation with ax² + bx + c = 0, we get a = 2, b = -3, and c = 8. So the discriminant is,
Δ OR D = b² − 4ac = (-3)² - 4(2)(8) = 9 - 64 = -55.

Discriminant of Cubic Equation

The discriminant of a cubic equation ax³ + bx² + cx + d = 0 is in terms of a, b, c, and d. i.e.,

• Δ or D = b²c² − 4ac³ − 4b³d − 27a²d² + 18abcd

Example: Find the discriminant of the cubic equation x³ - 3x + 2 = 0.

Comparing the equation with ax³ + bx² + cx + d = 0, we have a = 1, b = 0, c = -3, and d = 2. So its discriminant is,

Δ or D = b²c² − 4ac³ − 4b³d − 27a²d² + 18abcd
= (0)²(-3)² − 4(1)(-3)³ − 4(0)³(2) − 27(1)²(2)² + 18(1)(0)(-3)(2)
= 0 + 108 - 0 - 108 + 0
= 0

Discriminant and Nature of the Roots

• The roots of a quadratic equation ax² + bx + c = 0 are the values of x that satisfy the equation. They can be found using the quadratic formula: x = Though we cannot find the roots by just using the discriminant, we can determine the nature of the roots as follows.

If Discriminant is Positive

• If D > 0, the quadratic equation has two different real roots. This is because, when D > 0, the roots are given by x = and the square root of a positive number always results in a real number. So when the discriminant of a quadratic equation is greater than 0, it has two roots which are distinct and real numbers.

If Discriminant is Negative

• If D < 0, the quadratic equation has two different complex roots. This is because, when D < 0, the roots are given by x =  and the square root of a negative number leads to an imaginary number always. For example √−4 = 2i. So when the discriminant of a quadratic equation is less than 0, it has two roots which are distinct and complex numbers (non-real).

If Discriminant is Equal to Zero

• If D = 0, the quadratic equation has two equal real roots. In other words, when D = 0, the quadratic equation has only one real root. This is because, when D = 0, the roots are given by x =  and the square root of a 0 is 0. Then the equation turns into x = -b/2a which is only one number. So when the discriminant of a quadratic equation is zero, it has only one real root.
• A root is nothing but the x-coordinate of the x-intercept of the quadratic function. The graph of a quadratic function in each of these 3 cases can be as follows.
  

Important Notes on Discriminant:

• The discriminant of a quadratic equation ax² + bx + c = 0 is Δ OR D = b² − 4ac.
• A quadratic equation with discriminant D has:
  • two unequal real roots when D > 0
  • only one real root when D = 0
  • no real roots or two complex roots when D < 0