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Consider the equation 9x2 + 12x - 5=0.
(a) Find and state the value of the discriminant, b2 – 4ac. Then state whether one real-number solution, two different real-number solutions, or two different imaginary-number solutions exists. Show work.
 
 
(b) Find the exact solutions of the equation. Simplify as much as possible. Show work. 
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1 Answer

Recall that the solution to a quadratic equation of the form: ax2+bx+c=0 is given by:
 
   x=[-b±√(b2-4ac)]/2a
 
Note that the discriminant is the value under the square root, hence is:
 
i) A real number if b2-4ac>0 (because we can find the square root of a positive number)
ii) Zero if the number b2-4ac=0  (because the square root of zero is zero)
iii) A complex number if b2-4ac<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 9x2+12x-5-0, we have that:
 
   a=9
   b=12
   c=-5
 
therefore the discriminant b2-4ac = 122-4(9)(-5)
                                              = 144+180
                                              = 324
                                              > 0   →  two distinct real roots
 
 
b) x=[-b±√(b2-4ac)]/2a
 
     =[-12±√(122+12(15))]/2(9)
 
     =[-12±√[12(12+15)]]/18
 
     =[-12±√[12(27)]]/18
 
     =[-12±√(4(34))]/18
 
     =[-12±2(32)]/18
 
     =[-12±18]/18
 
     ={(-12+18)/18, (-12-18)/18}
 
     ={6/18, -30,18}
 
     ={1/3, -5/3}