5x^2+7x+3=0

Question

find the exact solutions to the quadratic equation in the complex numbers 

Answer 1

ax² + bx + c = 0
x₁₂ = (-b ± √D) / 2a , D = b² - 4ac .
5x² + 7x + 3 = 0
a = 5 , b = 7 , c = 3
D = 49 - 60 = - 11 , x₁ = (-7 - i √11) / 10
                             x₂ = (-7 + i √11) / 10  

Answer 2

We can find the answer by either using the quadratic equation or by completing the square.  I am going to complete the square.

5x² + 7x + 3 = 0                                   Given

5x² + 7x = -3                                       Subtract 3 from each side

x² + (7/5)x = -3/5                                Divide each term by 5

x² + (7/5)x + 49/100 = -3/5 + 49/100    Add 49/100 to each side ([(7/5)/2]² = 49/100)

(x + 7/10)² = -3/5 + 49/100                  Factor the left side to get a perfect square

(x + 7/10)² = -60/100 + 49/100             Find the common denominator on the right side

(x + 7/10)² = -11/100                           Simplify the right side

x + 7/10 = +/-√(-11/100)                      Take the square root of each side (remember the +/-)

x + 7/10 = +/-(√11/10)i                        Simplify the right side

x = -7/10 +/-(√11/10)i                          Subtract 7/10 from each side