A quadratic equation can be solved using the quadratic formula, provided

that a solution exists.

The quadratic equation

ax^{2} + bx + c = 0

where a ≠ 0, has up to two solutions given by

The part under the square root, b^{2} - 4ac, is called the discriminant and must be positive or zero for it to be "square rootable".

So....

In some cases, there will be no real solution and this occurs when the expression inside the square root, the discriminant, is negative.

i.e. when b^{2} - 4ac < 0

In some cases, there will be only one real solution and this occurs when the expression inside the square root, the discriminant, is zero.

i.e. when b^{2} - 4ac = 0

In all other cases, there will be two real solutions and this occurs when the expression inside the square root, the discriminant, is positive.

i.e. when b^{2} - 4ac > 0

**Example**

State the number of solutions to:

3x^{2} - 7x + 5 = 0

**Answer**

a = 3, b = - 7, c = 5

b^{2} = 49

4ac = 4 x 3 x 5 = 60

So the discriminant, b^{2} - 4ac < 0

**i.e. No real solutions.**