# The Discriminant | Revision

The discriminant is used to work out if a quadratic equation (ax2 +bx + c) has either any real roots or real solutions. To work out the discriminant, you use the value: b2 – 4ac.

Rearrange Example:

ax2 +bx + c = 4x2 +2x + 3

b2 – 4ac = 22 – 4 x 4 x 3

There are three possible conditions for the discriminant:

First condition: b2 -4ac > 0 means two distinct real roots:

The graph below: Y = x+ 4x + 2

Discriminant = 42 – 4 x 1 x 2 = 8

= 8 > 0

Two distinct roots

Second condition: b2 -4ac = 0 means two equal real roots/one repeated real root:

The graph below: Y = x– 6x + 9

(-6) 2 – 4 x 1 x 9 = 0

= 0 = 0

Two equal real roots/one repeated real root

Third condition b2 -4ac < 0 means no real roots:

The graph below: Y = 2x– x + 3

(-1) 2 – 4 x 2 x 3

= -23 < 0

No real roots

Solving Quadratic Equations Using the Discriminant:

Example:

The equation x2  + 4qx + 2q = 0, where q is a non-zero constant, has equal roots.

Find the value of q.

It is said that q is a non-zero, meaning that the correct solution is ½

The equation must be in the form ax2 +bx + c before you work out the values of a, b and c:

The equation 2x2 – kx + 6 = k has no real solutions for x.

Show that k2 + 8k – 48 < 0

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