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# Equation help

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.

### 1 Answer by Expert Tutors

Kahroline D. | I CAN teach you Mathematics!!I CAN teach you Mathematics!!
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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}