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Identical expressions
- (6 two 37x^2+411x+ two hundred and fifty)/ six thousand, two hundred and thirty-seven = zero
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- (6237x2+411x+250)/6237=0
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Similar expressions
- (6237x^2-411x+250)/6237=0
- (6237x^2+411x-250)/6237=0
(6237x^2+411x+250)/6237=0 equation
The solution
You have entered
[src]
2
6237*x + 411*x + 250
--------------------- = 0
6237
$$\frac{\left(6237 x^{2} + 411 x\right) + 250}{6237} = 0$$
Detail solution
Expand the expression in the equation
$$\frac{\left(6237 x^{2} + 411 x\right) + 250}{6237} = 0$$
We get the quadratic equation
$$x^{2} + \frac{137 x}{2079} + \frac{250}{6237} = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = \frac{137}{2079}$$
$$c = \frac{250}{6237}$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \frac{137}{4158} + \frac{\sqrt{674231} i}{4158}$$
$$x_{2} = - \frac{137}{4158} - \frac{\sqrt{674231} i}{4158}$$
$$\frac{\left(6237 x^{2} + 411 x\right) + 250}{6237} = 0$$
We get the quadratic equation
$$x^{2} + \frac{137 x}{2079} + \frac{250}{6237} = 0$$
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:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = \frac{137}{2079}$$
$$c = \frac{250}{6237}$$
, then
D = b^2 - 4 * a * c =
(137/2079)^2 - 4 * (1) * (250/6237) = -674231/4322241
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
$$x_{1} = - \frac{137}{4158} + \frac{\sqrt{674231} i}{4158}$$
$$x_{2} = - \frac{137}{4158} - \frac{\sqrt{674231} i}{4158}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{137}{2079}$$
$$q = \frac{c}{a}$$
$$q = \frac{250}{6237}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{137}{2079}$$
$$x_{1} x_{2} = \frac{250}{6237}$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{137}{2079}$$
$$q = \frac{c}{a}$$
$$q = \frac{250}{6237}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{137}{2079}$$
$$x_{1} x_{2} = \frac{250}{6237}$$
Sum and product of roots
[src]
sum
________ ________ 137 I*\/ 674231 137 I*\/ 674231 - ---- - ------------ + - ---- + ------------ 4158 4158 4158 4158
$$\left(- \frac{137}{4158} - \frac{\sqrt{674231} i}{4158}\right) + \left(- \frac{137}{4158} + \frac{\sqrt{674231} i}{4158}\right)$$
=
-137 ----- 2079
$$- \frac{137}{2079}$$
product
/ ________\ / ________\ | 137 I*\/ 674231 | | 137 I*\/ 674231 | |- ---- - ------------|*|- ---- + ------------| \ 4158 4158 / \ 4158 4158 /
$$\left(- \frac{137}{4158} - \frac{\sqrt{674231} i}{4158}\right) \left(- \frac{137}{4158} + \frac{\sqrt{674231} i}{4158}\right)$$
=
250 ---- 6237
$$\frac{250}{6237}$$
250/6237
Rapid solution
[src]
________
137 I*\/ 674231
x1 = - ---- - ------------
4158 4158
$$x_{1} = - \frac{137}{4158} - \frac{\sqrt{674231} i}{4158}$$
________
137 I*\/ 674231
x2 = - ---- + ------------
4158 4158
$$x_{2} = - \frac{137}{4158} + \frac{\sqrt{674231} i}{4158}$$
x2 = -137/4158 + sqrt(674231)*i/4158
Numerical answer
[src]
x1 = -0.0329485329485329 - 0.197478524384922*i
x2 = -0.0329485329485329 + 0.197478524384922*i
x2 = -0.0329485329485329 + 0.197478524384922*i