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- Equation:
- Equation (10*sqrt(x^2-x-1)+3)/((sqrt(x))^2-x-1)=13
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- Sum of series:
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13
- Canonical form:
- 13
- Derivative of:
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13
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Identical expressions
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- (10*sqrt(x to the power of 2-x-1)+3)/((sqrt(x)) to the power of 2-x-1)=13
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- 10sqrtx2-x-1+3/sqrtx2-x-1=13
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Similar expressions
- (10*sqrt(x^2+x-1)+3)/((sqrt(x))^2-x-1)=13
- (10*sqrt(x^2-x-1)-3)/((sqrt(x))^2-x-1)=13
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(10*sqrt(x^2-x-1)+3)/((sqrt(x))^2-x-1)=13 equation
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____________
/ 2
10*\/ x - x - 1 + 3
---------------------- = 13
2
___
\/ x - x - 1
$$\frac{10 \sqrt{\left(x^{2} - x\right) - 1} + 3}{\left(\left(\sqrt{x}\right)^{2} - x\right) - 1} = 13$$
Detail solution
Given the equation
$$\frac{10 \sqrt{\left(x^{2} - x\right) - 1} + 3}{\left(\left(\sqrt{x}\right)^{2} - x\right) - 1} = 13$$
$$- 10 \sqrt{x^{2} - x - 1} = 16$$
We raise the equation sides to 2-th degree
$$100 x^{2} - 100 x - 100 = 256$$
$$100 x^{2} - 100 x - 100 = 256$$
Transfer the right side of the equation left part with negative sign
$$100 x^{2} - 100 x - 356 = 0$$
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
$$a = 100$$
$$b = -100$$
$$c = -356$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{1}{2} + \frac{\sqrt{381}}{10}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{381}}{10}$$
Because
$$\sqrt{x^{2} - x - 1} = - \frac{8}{5}$$
and
$$\sqrt{x^{2} - x - 1} \geq 0$$
then
$$- \frac{8}{5} \geq 0$$
The final answer:
This equation has no roots
$$\frac{10 \sqrt{\left(x^{2} - x\right) - 1} + 3}{\left(\left(\sqrt{x}\right)^{2} - x\right) - 1} = 13$$
$$- 10 \sqrt{x^{2} - x - 1} = 16$$
We raise the equation sides to 2-th degree
$$100 x^{2} - 100 x - 100 = 256$$
$$100 x^{2} - 100 x - 100 = 256$$
Transfer the right side of the equation left part with negative sign
$$100 x^{2} - 100 x - 356 = 0$$
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
$$a = 100$$
$$b = -100$$
$$c = -356$$
, then
D = b^2 - 4 * a * c =
(-100)^2 - 4 * (100) * (-356) = 152400
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{1}{2} + \frac{\sqrt{381}}{10}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{381}}{10}$$
Because
$$\sqrt{x^{2} - x - 1} = - \frac{8}{5}$$
and
$$\sqrt{x^{2} - x - 1} \geq 0$$
then
$$- \frac{8}{5} \geq 0$$
The final answer:
This equation has no roots