Discriminant


Quick recap of Quadratic formula

A quadratic equation is an equation of the form ax2 + bx + c = 0 where a, b, and c are constants and a ≠ 0

For any such quadratic equation, quadratic formula is:

Roots(x1, x2) = −b ± √   b2 − 4ac 
         2a          


Discriminant explained

In the quadratic formula above, the quantity "b2 - 4ac" which is under the square root sign is called the discriminant of the quadratic equation.

Discriminant = b2 - 4ac

Example to calculate discriminant

For a quadratic equation x2 + 4x + 4 = 0, determine the determinant

Answer

x2 + 5x + 5 = 0

Discriminant = b2 – 4ac

Here, a = 1, b = 5, c = 5

Discriminant = b2 - 4ac = 52 - 4(1)(5) = 5

Use of discriminant

Discriminant tells about the nature of the roots of the quadratic equation. The roots may be real, equal or imaginary.

Depending upon the value of discriminant, a quadratic equation could have :

◾ two real and unequal roots when discriminant value > 0

◾ only one real and equal root when discriminant value = 0

◾ no real roots ie. both the roots are imaginary when discriminant value < 0

Case I- When Discriminant > 0

For a quadratic equation x2 - 12x + 32 = 0, determine the discriminant and the roots of the quadratic equation

Answer

Discriminant = b2 – 4ac

Here, a = 1, b = -12 , c = 32

Discriminant = (-12)2 - 4(1)(32) = 144 - 128 = 16

We know that the quadratic formula is

Roots(x1, x2) = −b ± √   b2 − 4ac 
         2a          


x1 = −(-12) + √   (-12)2 − 4(1)(32)  = 8
         2(1)          
x1 = −(-12) - √   (-12)2 - 4(1)(32)  = 4
         2(1)          

Hence, for the quadratic equation x2 - 12x + 32 = 0, we get two real and distinct roots- 8 and 4

Hence, for Discriminant > 0, the quadratic equation has 2 real and distinct roots



Case II- When Discriminant = 0

For a quadratic equation x2 - 12x + 36 = 0, determine the discriminant and the roots of the quadratic equation

Answer

Discriminant = b2 – 4ac

Here, a = 1, b = -12 , c = 36

Discriminant = (-12)2 - 4(1)(36) = 144 - 144 = 0

We know that the quadratic formula is

Roots(x1, x2) = −b ± √   b2 − 4ac 
         2a          


x1 = −(-12) + √   (-12)2 − 4(1)(36)  = 6
         2(1)          
x1 = −(-12) - √   (-12)2 - 4(1)(36)  = 6
         2(1)          

Hence, for the quadratic equation x2 - 12x + 36 = 0, we get only one real and equal root, which is 6.

Hence, for Discriminant = 0, the quadratic equation has only one real and equal root



Case III- When Discriminant < 0

For a quadratic equation x2 + 4x + 10 = 0, determine the discriminant and the roots of the quadratic equation

Answer

Discriminant = b2 – 4ac

Here, a = 1, b = 4 , c = 10

Discriminant = (4)2 - 4(1)(10) = 16 - 40 = -24

We know that the quadratic formula is

Roots(x1, x2) = −b ± √   b2 − 4ac 
         2a          


x1 = −(4) + √(4)2 − 4(1)(10)  = −(4) + √ -24
         2(1)                    2         
x1 = −(4) - √(4)2 - 4(1)(10)  = −(4) - √-24
         2(1)                    2         

Hence, for the quadratic equation x2 + 4x + 10 = 0, we get no real roots

Hence, for Discriminant < 0, the quadratic equation has no real roots



Summary

Discriminant value Cases Roots of quadratic Factorisation of quadratic
Discriminant value > 0 two real distinct roots two distinct linear factors
Discriminant value = 0 two identical real roots two identical linear factors
Discriminant value < 0 No real roots Unable to factorise

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