Question 59581
(Not necessary to find the roots - just determine the number & type of solution) use a discriminant to determine the number of solutions for the quadratic equasions - and if the solutions are real or complex.
:
The discriminant of a quadratic equation written in standard form {{{ax^2+bx+c=0}}} is {{{highlight(b^2-4ac)}}}
If the discriminant is positive, there are two real solutions.
If the discriminant is 0, there is one real solution.
If the discriminant is negative, there are two complex solutions.
:
a. 2x^2+x-1=0  a=2, b=1, c=-1
{{{b^2-4ac=(1)^2-4(2)(-1)}}}
={{{1+8=9}}}  The discriminant is positive, there are two real solutions.
:

b. 4/3x^2-2x+3/4=0  a=4/3, b=-2, c=3/4
{{{b^2-4ac=(-2)^2-4(4/3)(3/4)}}}
={{{4-4(12/12))}}}
{{{4-4(1)}}}
{{{4-4}}}
{{{0}}}  The discriminant is 0, there is one real solution.
:
c. m^2+m+1=0  a=1, b=1, c=1
{{{b^2-4ac=(1)^2-4(1)(1)}}}
={{{1-4}}}
={{{-3}}}  The discriminant is negative, there are two complex solutions.
:
Happy Calculating!!!