Inequality Proofs & Discriminants Revision Notes for Leaving Cert Mathematics

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 $k>3$ .

Inequality Proofs & Discriminants (Leaving Cert Mathematics): Revision Notes

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.

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Inequalities

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