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

1 Answer
Feb 13, 2017

Answer:

#368 >0# and not a perfect square

Therefore, the roots will be real and unequal

Explanation:

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