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(-3x^2+6x+16)=0

(-3x^2+6x+16)=0 equation

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Numerical solution:

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The solution

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     2               
- 3*x  + 6*x + 16 = 0
−3x2+6x+16=0- 3 x^{2} + 6 x + 16 = 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=6b = 6
c=16c = 16
, then
D=b2−4 a c=D = b^2 - 4\ a\ c =
62−(−3)4⋅16=2286^{2} - \left(-3\right) 4 \cdot 16 = 228
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=−573+1x_{1} = - \frac{\sqrt{57}}{3} + 1
Simplify
x2=1+573x_{2} = 1 + \frac{\sqrt{57}}{3}
Simplify
Vieta's Theorem
rewrite the equation
−3x2+6x+16=0- 3 x^{2} + 6 x + 16 = 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
x2−2x−163=0x^{2} - 2 x - \frac{16}{3} = 0
px+x2+q=0p x + x^{2} + q = 0
where
p=bap = \frac{b}{a}
p=−2p = -2
q=caq = \frac{c}{a}
q=−163q = - \frac{16}{3}
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=2x_{1} + x_{2} = 2
x1x2=−163x_{1} x_{2} = - \frac{16}{3}
The graph
05-15-10-5101520-200100
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
      ____         ____
    \/ 57        \/ 57 
1 - ------ + 1 + ------
      3            3
(−573+1)+(1+573)\left(- \frac{\sqrt{57}}{3} + 1\right) + \left(1 + \frac{\sqrt{57}}{3}\right) 
=
2
22
Simplify 
product
      ____         ____
    \/ 57        \/ 57 
1 - ------ * 1 + ------
      3            3
(−573+1)∗(1+573)\left(- \frac{\sqrt{57}}{3} + 1\right) * \left(1 + \frac{\sqrt{57}}{3}\right) 
=
-16/3
−163- \frac{16}{3}
Simplify
Rapid solution [src] 
            ____
          \/ 57 
x_1 = 1 - ------
            3
x1=−573+1x_{1} = - \frac{\sqrt{57}}{3} + 1
Simplify 
            ____
          \/ 57 
x_2 = 1 + ------
            3
x2=1+573x_{2} = 1 + \frac{\sqrt{57}}{3}
Simplify
Numerical answer [src] 
x1 = 3.51661147842358
x2 = -1.51661147842358
x2 = -1.51661147842358
The graph
(-3x^2+6x+16)=0 equation
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