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-5x^2-9x+12=0 equation

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Numerical solution:

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The solution

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     2               
- 5*x  - 9*x + 12 = 0
(−5x2−9x)+12=0\left(- 5 x^{2} - 9 x\right) + 12 = 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=−9b = -9
c=12c = 12
, then
D = b^2 - 4 * a * c = 

(-9)^2 - 4 * (-5) * (12) = 321

Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

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

or
x1=−32110−910x_{1} = - \frac{\sqrt{321}}{10} - \frac{9}{10}
x2=−910+32110x_{2} = - \frac{9}{10} + \frac{\sqrt{321}}{10}
Vieta's Theorem
rewrite the equation
(−5x2−9x)+12=0\left(- 5 x^{2} - 9 x\right) + 12 = 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+9x5−125=0x^{2} + \frac{9 x}{5} - \frac{12}{5} = 0
px+q+x2=0p x + q + x^{2} = 0
where
p=bap = \frac{b}{a}
p=95p = \frac{9}{5}
q=caq = \frac{c}{a}
q=−125q = - \frac{12}{5}
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=−95x_{1} + x_{2} = - \frac{9}{5}
x1x2=−125x_{1} x_{2} = - \frac{12}{5}
The graph
05-15-10-51015-10001000
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
              _____
       9    \/ 321 
x1 = - -- + -------
       10      10
x1=−910+32110x_{1} = - \frac{9}{10} + \frac{\sqrt{321}}{10} 
              _____
       9    \/ 321 
x2 = - -- - -------
       10      10
x2=−32110−910x_{2} = - \frac{\sqrt{321}}{10} - \frac{9}{10} 
x2 = -sqrt(321)/10 - 9/10
Sum and product of roots [src] 
sum
         _____            _____
  9    \/ 321      9    \/ 321 
- -- + ------- + - -- - -------
  10      10       10      10
(−32110−910)+(−910+32110)\left(- \frac{\sqrt{321}}{10} - \frac{9}{10}\right) + \left(- \frac{9}{10} + \frac{\sqrt{321}}{10}\right) 
=
-9/5
−95- \frac{9}{5} 
product
/         _____\ /         _____\
|  9    \/ 321 | |  9    \/ 321 |
|- -- + -------|*|- -- - -------|
\  10      10  / \  10      10  /
(−910+32110)(−32110−910)\left(- \frac{9}{10} + \frac{\sqrt{321}}{10}\right) \left(- \frac{\sqrt{321}}{10} - \frac{9}{10}\right) 
=
-12/5
−125- \frac{12}{5} 
-12/5
Numerical answer [src] 
x1 = 0.891647286716892
x2 = -2.69164728671689
x2 = -2.69164728671689
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