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Similar expressions
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x^2+6/5-8-x/10=0 equation
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The solution
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[src]
2 6 x
x + - - 8 - -- = 0
5 10
$$- \frac{x}{10} + \left(\left(x^{2} + \frac{6}{5}\right) - 8\right) = 0$$
Detail solution
Expand the expression in the equation
$$- \frac{x}{10} + \left(\left(x^{2} + \frac{6}{5}\right) - 8\right) = 0$$
We get the quadratic equation
$$x^{2} - \frac{x}{10} - \frac{34}{5} = 0$$
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
$$a = 1$$
$$b = - \frac{1}{10}$$
$$c = - \frac{34}{5}$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{1}{20} + \frac{\sqrt{2721}}{20}$$
$$x_{2} = \frac{1}{20} - \frac{\sqrt{2721}}{20}$$
$$- \frac{x}{10} + \left(\left(x^{2} + \frac{6}{5}\right) - 8\right) = 0$$
We get the quadratic equation
$$x^{2} - \frac{x}{10} - \frac{34}{5} = 0$$
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
$$a = 1$$
$$b = - \frac{1}{10}$$
$$c = - \frac{34}{5}$$
, then
D = b^2 - 4 * a * c =
(-1/10)^2 - 4 * (1) * (-34/5) = 2721/100
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{1}{20} + \frac{\sqrt{2721}}{20}$$
$$x_{2} = \frac{1}{20} - \frac{\sqrt{2721}}{20}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{1}{10}$$
$$q = \frac{c}{a}$$
$$q = - \frac{34}{5}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{1}{10}$$
$$x_{1} x_{2} = - \frac{34}{5}$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{1}{10}$$
$$q = \frac{c}{a}$$
$$q = - \frac{34}{5}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{1}{10}$$
$$x_{1} x_{2} = - \frac{34}{5}$$
Sum and product of roots
[src]
sum
______ ______ 1 \/ 2721 1 \/ 2721 -- - -------- + -- + -------- 20 20 20 20
$$\left(\frac{1}{20} - \frac{\sqrt{2721}}{20}\right) + \left(\frac{1}{20} + \frac{\sqrt{2721}}{20}\right)$$
=
1/10
$$\frac{1}{10}$$
product
/ ______\ / ______\ |1 \/ 2721 | |1 \/ 2721 | |-- - --------|*|-- + --------| \20 20 / \20 20 /
$$\left(\frac{1}{20} - \frac{\sqrt{2721}}{20}\right) \left(\frac{1}{20} + \frac{\sqrt{2721}}{20}\right)$$
=
-34/5
$$- \frac{34}{5}$$
-34/5
Rapid solution
[src]
______
1 \/ 2721
x1 = -- - --------
20 20
$$x_{1} = \frac{1}{20} - \frac{\sqrt{2721}}{20}$$
______
1 \/ 2721
x2 = -- + --------
20 20
$$x_{2} = \frac{1}{20} + \frac{\sqrt{2721}}{20}$$
x2 = 1/20 + sqrt(2721)/20