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Similar expressions
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1/(x-1)^2+2/(x-1)-3=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
1 2
-------- + ----- - 3 = 0
2 x - 1
(x - 1)
$$\left(\frac{1}{\left(x - 1\right)^{2}} + \frac{2}{x - 1}\right) - 3 = 0$$
Detail solution
Given the equation:
$$\left(\frac{1}{\left(x - 1\right)^{2}} + \frac{2}{x - 1}\right) - 3 = 0$$
Multiply the equation sides by the denominators:
(-1 + x)^2
we get:
$$\left(x - 1\right)^{2} \left(\left(\frac{1}{\left(x - 1\right)^{2}} + \frac{2}{x - 1}\right) - 3\right) = 0$$
$$- 3 x^{2} + 8 x - 4 = 0$$
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
$$a = -3$$
$$b = 8$$
$$c = -4$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{2}{3}$$
$$x_{2} = 2$$
$$\left(\frac{1}{\left(x - 1\right)^{2}} + \frac{2}{x - 1}\right) - 3 = 0$$
Multiply the equation sides by the denominators:
(-1 + x)^2
we get:
$$\left(x - 1\right)^{2} \left(\left(\frac{1}{\left(x - 1\right)^{2}} + \frac{2}{x - 1}\right) - 3\right) = 0$$
$$- 3 x^{2} + 8 x - 4 = 0$$
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
$$a = -3$$
$$b = 8$$
$$c = -4$$
, then
D = b^2 - 4 * a * c =
(8)^2 - 4 * (-3) * (-4) = 16
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{2}{3}$$
$$x_{2} = 2$$
Sum and product of roots
[src]
sum
2 + 2/3
$$\frac{2}{3} + 2$$
=
8/3
$$\frac{8}{3}$$
product
2*2 --- 3
$$\frac{2 \cdot 2}{3}$$
=
4/3
$$\frac{4}{3}$$
4/3
The graph