How do you use the discriminant to determine the nature of the solutions given `4x^2 + 12x = 14`?

When you are solving a quadratic equation, it is very useful to know what sort of answer you will get. This can often help in determining which method to use - for example whether to look for factors or to use the quadratic formula.

A quadratic equation is written in the form `ax^2 +bx +c =0`
Always change to this form first

The discriminant is `Delta = b^2-4ac`
The solutions to an equation are called the 'roots' and are referred to as `alpha and beta`

The value of `Delta` tells us about the nature of the roots.

If `Delta > 0 rArr` the roots are real and unequal (2 distinct roots)

If `Delta > 0 " and a prefect square" rArr` the roots are real, unequal and`color(white)(.................................. .................)` rational

If `Delta = 0 rArr` the roots are real and equal (1 root)

If `Delta < 0 rArr` the roots are imaginary and unequal

Note that if `a " or " b` are irrational, the roots will be irrational.

`4x^2 +12x = 14 " "rArr" 4x^2 +12x -14 =0`

`Delta = b^2 -4ac`

`Delta = (12)^2 -4(4)(-14) = 368`

`368 >0` and not a perfect square

Therefore, the roots will be real and unequal