Mister Exam

Other calculators

5x^2+3x+6=0

5x^2+3x+6=0 equation

The teacher will be very surprised to see your correct solution 😉

v

Numerical solution:

Do search numerical solution at [, ]

The solution

You have entered [src] 
   2              
5*x  + 3*x + 6 = 0
5x2+3x+6=05 x^{2} + 3 x + 6 = 0
Simplify
Detail solution
This equation is of the form
a*x^2 + b*x + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
where D = b^2 - 4*a*c - it is the discriminant.
Because
a=5a = 5
b=3b = 3
c=6c = 6
, then
D = b^2 - 4 * a * c = 

(3)^2 - 4 * (5) * (6) = -111

Because D<0, then the equation
has no real roots,
but complex roots is exists.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
x1=−310+111i10x_{1} = - \frac{3}{10} + \frac{\sqrt{111} i}{10}
Simplify
x2=−310−111i10x_{2} = - \frac{3}{10} - \frac{\sqrt{111} i}{10}
Simplify
Vieta's Theorem
rewrite the equation
5x2+3x+6=05 x^{2} + 3 x + 6 = 0
of
ax2+bx+c=0a x^{2} + b x + c = 0
as reduced quadratic equation
x2+bxa+ca=0x^{2} + \frac{b x}{a} + \frac{c}{a} = 0
x2+3x5+65=0x^{2} + \frac{3 x}{5} + \frac{6}{5} = 0
px+x2+q=0p x + x^{2} + q = 0
where
p=bap = \frac{b}{a}
p=35p = \frac{3}{5}
q=caq = \frac{c}{a}
q=65q = \frac{6}{5}
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=−35x_{1} + x_{2} = - \frac{3}{5}
x1x2=65x_{1} x_{2} = \frac{6}{5}
The graph
-2.5-2.0-1.5-1.0-0.50.00.51.01.5020
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
                _____
       3    I*\/ 111 
x1 = - -- - ---------
       10       10
x1=−310−111i10x_{1} = - \frac{3}{10} - \frac{\sqrt{111} i}{10}
Simplify 
                _____
       3    I*\/ 111 
x2 = - -- + ---------
       10       10
x2=−310+111i10x_{2} = - \frac{3}{10} + \frac{\sqrt{111} i}{10}
Simplify
Sum and product of roots [src] 
sum
               _____              _____
      3    I*\/ 111      3    I*\/ 111 
0 + - -- - --------- + - -- + ---------
      10       10        10       10
(0−(310+111i10))−(310−111i10)\left(0 - \left(\frac{3}{10} + \frac{\sqrt{111} i}{10}\right)\right) - \left(\frac{3}{10} - \frac{\sqrt{111} i}{10}\right) 
=
-3/5
−35- \frac{3}{5} 
product
  /           _____\ /           _____\
  |  3    I*\/ 111 | |  3    I*\/ 111 |
1*|- -- - ---------|*|- -- + ---------|
  \  10       10   / \  10       10   /
1(−310−111i10)(−310+111i10)1 \left(- \frac{3}{10} - \frac{\sqrt{111} i}{10}\right) \left(- \frac{3}{10} + \frac{\sqrt{111} i}{10}\right) 
=
6/5
65\frac{6}{5} 
6/5
Numerical answer [src] 
x1 = -0.3 - 1.05356537528527*i
x2 = -0.3 + 1.05356537528527*i
x2 = -0.3 + 1.05356537528527*i
The graph
5x^2+3x+6=0 equation
    © Mister Exam – Calculator