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Discriminant

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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.
Standard form of a quadratic equation is $ax^{2}$ + $bx$ + $c$ = 0
Discriminant value of a quadratic equation is : $b^{2}$ - 4$ac$. For a polynomial discriminant is a function of its coefficients and is denoted by a capital  letter "D" or a greek letter Delta( $\Delta$ ). The term discriminant was coined by British mathematician James Joseph Sylvester.

Formula

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To find the discriminant of a 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.

Examples

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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.