Discriminants

The discriminant is a key concept in the study of quadratic equations, typically of the form:

ax^2 + bx + c = 0

where $a,b,c$ are constants, and $a \neq 0$ . The discriminant is a part of the quadratic formula located underneath the square root.

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

D=b^2-4ac

The discriminant tells us about the nature of the roots of the quadratic equation:

Case 1

D > 0 :

The quadratic equation has two distinct real roots.
The roots are unequal and real.

Case 2

D = 0 :

The quadratic equation has one real root (or two equal real roots).
The root is real and repeated.

Case 3

D < 0 :

The quadratic equation has no real roots.
The roots are complex (conjugate pairs).

Example

infoNote

Determine the nature of the roots in the following quadratic equation : $2x^2+3x-2=0$ .

First identify the coefficients of the quadratic :

a=2,b=3,c=-2

Plug into the discriminant :

D=b^2-4ac=3^2-4(2)(-2)=25

Since $D>0$ , this quadratic has two distinct real roots.

Example

infoNote

Determine the nature of the roots in the following quadratic equation : $x^2+2x+5=0$ .

First identify the coefficients of the quadratic :

a=1,b=2,c=5

Plug into the discriminant :

D=b^2-4ac=2^2 - 4(1)(5) = -16

Since $D<0$ , this quadratic has two complex roots.

Example

infoNote

Given the quadratic equation $\ 3x^2 + 2kx + k = 0$ , find the range of values of $k$ for which the equation has two distinct real roots.

First identify the coefficients of the quadratic :

a=3,b=2k,c=k

If the quadratic has two distinct real roots, then $D>0$ .

b^2-4ac>0

(2k)^2-4(3)(k)>0

4k^2 - 12k >0

Solve the quadratic inequality :

4k(k-3)>0

The equation has two distinct real roots for $k<0$ or $k>3$ .

Discriminants Simplified Revision Notes for Leaving Cert Mathematics

Discriminants

Case 1

Case 2

Case 3

Determine the nature of the roots in the following quadratic equation : $2x^2+3x-2=0$ .

Determine the nature of the roots in the following quadratic equation : $x^2+2x+5=0$ .

Given the quadratic equation $\ 3x^2 + 2kx + k = 0$ , find the range of values of $k$ for which the equation has two distinct real roots.

500K+ Students Use These Powerful Tools to Master Discriminants For their Leaving Cert Exams.

Other Revision Notes related to Discriminants you should explore

Discriminants Simplified Revision Notes for Leaving Cert Mathematics

Discriminants

Case 1

Case 2

Case 3

Determine the nature of the roots in the following quadratic equation : 2x2+3x−2=02x^2+3x-2=02x2+3x−2=0.

Determine the nature of the roots in the following quadratic equation : x2+2x+5=0x^2+2x+5=0x2+2x+5=0.

Given the quadratic equation 3x2+2kx+k=0\ 3x^2 + 2kx + k = 0 3x2+2kx+k=0, find the range of values of kkk for which the equation has two distinct real roots.

500K+ Students Use These Powerful Tools to Master Discriminants For their Leaving Cert Exams.

Other Revision Notes related to Discriminants you should explore

Determine the nature of the roots in the following quadratic equation : $2x^2+3x-2=0$ .

Determine the nature of the roots in the following quadratic equation : $x^2+2x+5=0$ .

Given the quadratic equation $\ 3x^2 + 2kx + k = 0$ , find the range of values of $k$ for which the equation has two distinct real roots.