Discriminant is the name given to the expression under the square root sign in the quadratic formula. It gives a clear idea of the number of x -intercepts related to a quadratic equation. Generally it finds the number of real roots of quadratic equation and also the nature of the roots of a quadratic equation. |

**quadratic equation**the formula is

$D$ = $b^{2}$ - 4$ac$

where a, b and c are real numbers.

$D$ : Discriminant

**With the help of discriminant we can easily identify what type of roots the equation has:**

Discriminant ($D$ = $b^{2}$ - 4$ac$) |
Roots |

$D$ < 0 | Two zeroes that are complex conjugate. |

$D$ = 0 | One real zero of multiplicity two. |

$D$ > 0 | Two distinct real zeroes. |

Discriminant for a cubic polynomial

$ax^{3}$ + $bx^{2}$ + $cx$ + $d$ is

$\Delta$ = $b^{2}$c$^{2}$ - 4$ac^{3}$ - 4$b^{3}$d - 27a$^{2}$d$^{2}$ + 18abcd

**A polynomial will have multiple roots in the complex numbers if its discriminant is zero:**

$\Delta$ > 0: the equation has 3 distinct real roots;

$\Delta$ < 0, the equation has 1 real root and 2 complex conjugate roots;

$\Delta$ = 0: at least 2 roots coincide, and they are all real.

**Example 1 :**What is the discriminant value of a quadratic equation 4x$^{2}$ - 5x + 2 and determine the number of real roots.

**Solution:**

__Step 1:__Given quadratic equation is 4x$^{2}$ - 5x + 2

__Step 2:__The above quadratic equation is of the form ax$^{2}$ + bx + c = 0.

Here, a = 4; b = -5; c = 2;

__Step 3:__Discriminant formula for the given equation is b$^{2}$ - 4ac.

b$^{2}$ – 4$ac$ = (-5)$^{2}$ - 4(4) (2)

= 25 - 32

= - 7 < 0

The discriminant value for a given equation is less than zero.

__Step 4:__Hence, the given quadratic equation has

**no real roots**.

**Example 2 :**What is the discriminant of the equation x$^{2}$ + 3x + 4?

Solution:

In the given equation, coefficients of a, b and c are

a = 1 ; b = 3 ; c = 4

**The formula for discriminant is:**

$\Delta$ = $b^{2}$ - 4$ac$

$\Delta$ = (3)$^{2}$ - 4(1)(4)

$\Delta$ = 9 - 16

$\Delta$ = - 7

Therefore the discriminant for the given equation is

**- 7.**