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x^2+x+lg(sinx)=1+lg(sinx) equação
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A solução
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[src]
2 x + x + log(sin(x)) = 1 + log(sin(x))
$$x^{2} + x + \log{\left(\sin{\left(x \right)} \right)} = \log{\left(\sin{\left(x \right)} \right)} + 1$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$x^{2} + x + \log{\left(\sin{\left(x \right)} \right)} = \log{\left(\sin{\left(x \right)} \right)} + 1$$
to
$$\left(- \log{\left(\sin{\left(x \right)} \right)} - 1\right) + \left(x^{2} + x + \log{\left(\sin{\left(x \right)} \right)}\right) = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 1$$
$$b = 1$$
$$c = -1$$
, then
$$D = b^2 - 4 * a * c = $$
$$1^{2} - 1 \cdot 4 \left(-1\right) = 5$$
Because D > 0, then the equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = - \frac{1}{2} + \frac{\sqrt{5}}{2}$$
Simplify
$$x_{2} = - \frac{\sqrt{5}}{2} - \frac{1}{2}$$
Simplify
left part with negative sign.
The equation is transformed from
$$x^{2} + x + \log{\left(\sin{\left(x \right)} \right)} = \log{\left(\sin{\left(x \right)} \right)} + 1$$
to
$$\left(- \log{\left(\sin{\left(x \right)} \right)} - 1\right) + \left(x^{2} + x + \log{\left(\sin{\left(x \right)} \right)}\right) = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 1$$
$$b = 1$$
$$c = -1$$
, then
$$D = b^2 - 4 * a * c = $$
$$1^{2} - 1 \cdot 4 \left(-1\right) = 5$$
Because D > 0, then the equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = - \frac{1}{2} + \frac{\sqrt{5}}{2}$$
Simplify
$$x_{2} = - \frac{\sqrt{5}}{2} - \frac{1}{2}$$
Simplify
Vieta's Theorem
it is reduced quadratic equation
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = -1$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -1$$
$$x_{1} x_{2} = -1$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = -1$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -1$$
$$x_{1} x_{2} = -1$$
Rapid solution
[src]
___
1 \/ 5
x_1 = - - + -----
2 2
$$x_{1} = - \frac{1}{2} + \frac{\sqrt{5}}{2}$$
___
1 \/ 5
x_2 = - - - -----
2 2
$$x_{2} = - \frac{\sqrt{5}}{2} - \frac{1}{2}$$
Sum and product of roots
[src]
sum
___ ___ 1 \/ 5 1 \/ 5 - - + ----- + - - - ----- 2 2 2 2
$$\left(- \frac{1}{2} + \frac{\sqrt{5}}{2}\right) + \left(- \frac{\sqrt{5}}{2} - \frac{1}{2}\right)$$
=
-1
$$-1$$
product
___ ___ 1 \/ 5 1 \/ 5 - - + ----- * - - - ----- 2 2 2 2
$$\left(- \frac{1}{2} + \frac{\sqrt{5}}{2}\right) * \left(- \frac{\sqrt{5}}{2} - \frac{1}{2}\right)$$
=
-1
$$-1$$
Gráfico