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