The discriminant is used to work out if a quadratic equation (ax^{2}** **+bx + c) has either any real roots or real solutions. To work out the discriminant, you use the value: b^{2}** **– 4ac.

**Rearrange Example:**

ax^{2}** **+bx + c = 4x^{2}** **+2x + 3

b^{2}** **– 4ac = 2^{2}** **– 4 x 4 x 3

There are three possible conditions for the discriminant:

**First condition: b**^{2}** -4ac > 0 means two distinct real roots:**

The graph below: Y = x** ^{2 }**+ 4x + 2

Discriminant
= 4** ^{2}** – 4 x 1 x 2 = 8

= 8 > 0

Two distinct roots

**Second condition:** **b**^{2}** -4ac = 0 means two equal real roots/one repeated real root:**

The graph below: Y = x** ^{2 }**– 6x + 9

(-6)^{
}** ^{2}** – 4 x 1 x 9 = 0

= 0 = 0

Two equal real roots/one repeated real root

**Third condition** **b**^{2}** -4ac < 0 means no real roots:**

The graph
below: Y = 2x** ^{2 }**– x + 3

(-1)^{
}** ^{2}** – 4 x 2 x 3

= -23 < 0

No real roots

**Solving Quadratic Equations Using the Discriminant:**

Example:

The
equation x** ^{2}** + 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 ax^{2}** **+bx + c before you work out the
values of a, b and c:

The
equation 2x** ^{2}** – kx + 6 = k has no real solutions for x.

Show that k** ^{2 }**+ 8k – 48 < 0

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