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Similar expressions
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1/x^2-3/x-4=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
1 3 -- - - - 4 = 0 2 x x
$$\left(\frac{1}{x^{2}} - \frac{3}{x}\right) - 4 = 0$$
Detail solution
Given the equation:
$$\left(\frac{1}{x^{2}} - \frac{3}{x}\right) - 4 = 0$$
Multiply the equation sides by the denominators:
x^2
we get:
$$x^{2} \left(\left(\frac{1}{x^{2}} - \frac{3}{x}\right) - 4\right) = 0$$
$$- 4 x^{2} - 3 x + 1 = 0$$
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
$$a = -4$$
$$b = -3$$
$$c = 1$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = -1$$
$$x_{2} = \frac{1}{4}$$
$$\left(\frac{1}{x^{2}} - \frac{3}{x}\right) - 4 = 0$$
Multiply the equation sides by the denominators:
x^2
we get:
$$x^{2} \left(\left(\frac{1}{x^{2}} - \frac{3}{x}\right) - 4\right) = 0$$
$$- 4 x^{2} - 3 x + 1 = 0$$
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
$$a = -4$$
$$b = -3$$
$$c = 1$$
, then
D = b^2 - 4 * a * c =
(-3)^2 - 4 * (-4) * (1) = 25
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = -1$$
$$x_{2} = \frac{1}{4}$$
Sum and product of roots
[src]
sum
-1 + 1/4
$$-1 + \frac{1}{4}$$
=
-3/4
$$- \frac{3}{4}$$
product
-1 --- 4
$$- \frac{1}{4}$$
=
-1/4
$$- \frac{1}{4}$$
-1/4
The graph