5x^2+3x+6=0 equation
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The solution
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:
x 1 = D − b 2 a x_{1} = \frac{\sqrt{D} - b}{2 a} x 1 = 2 a D − b x 2 = − D − b 2 a x_{2} = \frac{- \sqrt{D} - b}{2 a} x 2 = 2 a − D − b where D = b^2 - 4*a*c - it is the discriminant.
Because
a = 5 a = 5 a = 5 b = 3 b = 3 b = 3 c = 6 c = 6 c = 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
x 1 = − 3 10 + 111 i 10 x_{1} = - \frac{3}{10} + \frac{\sqrt{111} i}{10} x 1 = − 10 3 + 10 111 i Simplify x 2 = − 3 10 − 111 i 10 x_{2} = - \frac{3}{10} - \frac{\sqrt{111} i}{10} x 2 = − 10 3 − 10 111 i Simplify
Vieta's Theorem
rewrite the equation
5 x 2 + 3 x + 6 = 0 5 x^{2} + 3 x + 6 = 0 5 x 2 + 3 x + 6 = 0 of
a x 2 + b x + c = 0 a x^{2} + b x + c = 0 a x 2 + b x + c = 0 as reduced quadratic equation
x 2 + b x a + c a = 0 x^{2} + \frac{b x}{a} + \frac{c}{a} = 0 x 2 + a b x + a c = 0 x 2 + 3 x 5 + 6 5 = 0 x^{2} + \frac{3 x}{5} + \frac{6}{5} = 0 x 2 + 5 3 x + 5 6 = 0 p x + x 2 + q = 0 p x + x^{2} + q = 0 p x + x 2 + q = 0 where
p = b a p = \frac{b}{a} p = a b p = 3 5 p = \frac{3}{5} p = 5 3 q = c a q = \frac{c}{a} q = a c q = 6 5 q = \frac{6}{5} q = 5 6 Vieta Formulas
x 1 + x 2 = − p x_{1} + x_{2} = - p x 1 + x 2 = − p x 1 x 2 = q x_{1} x_{2} = q x 1 x 2 = q x 1 + x 2 = − 3 5 x_{1} + x_{2} = - \frac{3}{5} x 1 + x 2 = − 5 3 x 1 x 2 = 6 5 x_{1} x_{2} = \frac{6}{5} x 1 x 2 = 5 6
The graph
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 0 20
_____
3 I*\/ 111
x1 = - -- - ---------
10 10
x 1 = − 3 10 − 111 i 10 x_{1} = - \frac{3}{10} - \frac{\sqrt{111} i}{10} x 1 = − 10 3 − 10 111 i
_____
3 I*\/ 111
x2 = - -- + ---------
10 10
x 2 = − 3 10 + 111 i 10 x_{2} = - \frac{3}{10} + \frac{\sqrt{111} i}{10} x 2 = − 10 3 + 10 111 i
Sum and product of roots
[src]
_____ _____
3 I*\/ 111 3 I*\/ 111
0 + - -- - --------- + - -- + ---------
10 10 10 10
( 0 − ( 3 10 + 111 i 10 ) ) − ( 3 10 − 111 i 10 ) \left(0 - \left(\frac{3}{10} + \frac{\sqrt{111} i}{10}\right)\right) - \left(\frac{3}{10} - \frac{\sqrt{111} i}{10}\right) ( 0 − ( 10 3 + 10 111 i ) ) − ( 10 3 − 10 111 i )
/ _____\ / _____\
| 3 I*\/ 111 | | 3 I*\/ 111 |
1*|- -- - ---------|*|- -- + ---------|
\ 10 10 / \ 10 10 /
1 ( − 3 10 − 111 i 10 ) ( − 3 10 + 111 i 10 ) 1 \left(- \frac{3}{10} - \frac{\sqrt{111} i}{10}\right) \left(- \frac{3}{10} + \frac{\sqrt{111} i}{10}\right) 1 ( − 10 3 − 10 111 i ) ( − 10 3 + 10 111 i )
x1 = -0.3 - 1.05356537528527*i
x2 = -0.3 + 1.05356537528527*i
x2 = -0.3 + 1.05356537528527*i