Recall that the solution to a quadratic equation of the form: ax^{2}+bx+c=0 is given by:

x=[-b±√(b^{2}-4ac)]/2a

Note that the discriminant is the value under the square root, hence is:

i) A real number if b^{2}-4ac>0 (because we can find the square root of a positive number)

ii) Zero if the number b^{2}-4ac=0 (because the square root of zero is zero)

iii) A complex number if b^{2}-4ac<0 (because the square root of a negative number has a factor of i)

Due to the ± preceding the square root in the quadratic equation, cases i) and iii) above will correspond to two distinct values for x

a) Given 9x^{2}+12x-5-0, we have that:

a=9

b=12

c=-5

therefore the discriminant b^{2}-4ac = 12^{2}-4(9)(-5)

= 144+180

= 324

> 0 → two distinct real roots

b) x=[-b±√(b^{2}-4ac)]/2a

=[-12±√(12^{2}+12(15))]/2(9)

=[-12±√[12(12+15)]]/18

=[-12±√[12(27)]]/18

=[-12±√(4(3^{4}))]/18

=[-12±2(3^{2})]/18

=[-12±18]/18

={(-12+18)/18, (-12-18)/18}

={6/18, -30,18}

={1/3, -5/3}