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x^2-4*x-6=0

x^2-4*x-6=0 equation

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Numerical solution:

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The solution

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 2              
x  - 4*x - 6 = 0
x2−4x−6=0x^{2} - 4 x - 6 = 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=1a = 1
b=−4b = -4
c=−6c = -6
, then
D=b2−4 a c=D = b^2 - 4\ a\ c =
(−4)2−1⋅4(−6)=40\left(-4\right)^{2} - 1 \cdot 4 \left(-6\right) = 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=2+10x_{1} = 2 + \sqrt{10}
Simplify
x2=−10+2x_{2} = - \sqrt{10} + 2
Simplify
Vieta's Theorem
it is reduced quadratic equation
px+x2+q=0p x + x^{2} + q = 0
where
p=bap = \frac{b}{a}
p=−4p = -4
q=caq = \frac{c}{a}
q=−6q = -6
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=4x_{1} + x_{2} = 4
x1x2=−6x_{1} x_{2} = -6
The graph
05-15-10-5101520200-100
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
            ____
x_1 = 2 - \/ 10
x1=−10+2x_{1} = - \sqrt{10} + 2
Simplify 
            ____
x_2 = 2 + \/ 10
x2=2+10x_{2} = 2 + \sqrt{10}
Simplify
Sum and product of roots [src] 
sum
      ____         ____
2 - \/ 10  + 2 + \/ 10
(−10+2)+(2+10)\left(- \sqrt{10} + 2\right) + \left(2 + \sqrt{10}\right) 
=
4
44
Simplify 
product
      ____         ____
2 - \/ 10  * 2 + \/ 10
(−10+2)∗(2+10)\left(- \sqrt{10} + 2\right) * \left(2 + \sqrt{10}\right) 
=
-6
−6-6
Simplify
Numerical answer [src] 
x1 = 5.16227766016838
x2 = -1.16227766016838
x2 = -1.16227766016838
The graph
x^2-4*x-6=0 equation
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