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4x-3x^2+2=0

4x-3x^2+2=0 equation

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Numerical solution:

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The solution

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         2        
4*x - 3*x  + 2 = 0
−3x2+4x+2=0- 3 x^{2} + 4 x + 2 = 0
Simplify
Detail solution
This equation is of the form
a x2+b x+c=0a\ 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=b2−4acD = b^2 - 4 a c is the discriminant.
Because
a=−3a = -3
b=4b = 4
c=2c = 2
, then
D=b2−4 a c=D = b^2 - 4\ a\ c =
42−(−3)4⋅2=404^{2} - \left(-3\right) 4 \cdot 2 = 40
Because D > 0, then the equation has two roots.
x1=(−b+D)2ax_1 = \frac{(-b + \sqrt{D})}{2 a}
x2=(−b−D)2ax_2 = \frac{(-b - \sqrt{D})}{2 a}
or
x1=−103+23x_{1} = - \frac{\sqrt{10}}{3} + \frac{2}{3}
Simplify
x2=23+103x_{2} = \frac{2}{3} + \frac{\sqrt{10}}{3}
Simplify
Vieta's Theorem
rewrite the equation
−3x2+4x+2=0- 3 x^{2} + 4 x + 2 = 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
x2−4x3−23=0x^{2} - \frac{4 x}{3} - \frac{2}{3} = 0
px+x2+q=0p x + x^{2} + q = 0
where
p=bap = \frac{b}{a}
p=−43p = - \frac{4}{3}
q=caq = \frac{c}{a}
q=−23q = - \frac{2}{3}
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=43x_{1} + x_{2} = \frac{4}{3}
x1x2=−23x_{1} x_{2} = - \frac{2}{3}
The graph
05-15-10-51015-100100
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
            ____
      2   \/ 10 
x_1 = - - ------
      3     3
x1=−103+23x_{1} = - \frac{\sqrt{10}}{3} + \frac{2}{3}
Simplify 
            ____
      2   \/ 10 
x_2 = - + ------
      3     3
x2=23+103x_{2} = \frac{2}{3} + \frac{\sqrt{10}}{3}
Simplify
Sum and product of roots [src] 
sum
      ____         ____
2   \/ 10    2   \/ 10 
- - ------ + - + ------
3     3      3     3
(−103+23)+(23+103)\left(- \frac{\sqrt{10}}{3} + \frac{2}{3}\right) + \left(\frac{2}{3} + \frac{\sqrt{10}}{3}\right) 
=
4/3
43\frac{4}{3}
Simplify 
product
      ____         ____
2   \/ 10    2   \/ 10 
- - ------ * - + ------
3     3      3     3
(−103+23)∗(23+103)\left(- \frac{\sqrt{10}}{3} + \frac{2}{3}\right) * \left(\frac{2}{3} + \frac{\sqrt{10}}{3}\right) 
=
-2/3
−23- \frac{2}{3}
Simplify
Numerical answer [src] 
x1 = 1.72075922005613
x2 = -0.387425886722793
x2 = -0.387425886722793
The graph
4x-3x^2+2=0 equation
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