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1/(x-4)2-(7/x-4)+10=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
2 7 ----- + - - + 4 + 10 = 0 x - 4 x
$$\left(\left(4 - \frac{7}{x}\right) + \frac{2}{x - 4}\right) + 10 = 0$$
Detail solution
Given the equation:
$$\left(\left(4 - \frac{7}{x}\right) + \frac{2}{x - 4}\right) + 10 = 0$$
Multiply the equation sides by the denominators:
x and -4 + x
we get:
$$x \left(\left(\left(4 - \frac{7}{x}\right) + \frac{2}{x - 4}\right) + 10\right) = 0$$
$$\frac{14 x^{2} - 61 x + 28}{x - 4} = 0$$
$$\frac{14 x^{2} - 61 x + 28}{x - 4} \left(x - 4\right) = 0 \left(x - 4\right)$$
$$14 x^{2} - 61 x + 28 = 0$$
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
$$a = 14$$
$$b = -61$$
$$c = 28$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{\sqrt{2153}}{28} + \frac{61}{28}$$
$$x_{2} = \frac{61}{28} - \frac{\sqrt{2153}}{28}$$
$$\left(\left(4 - \frac{7}{x}\right) + \frac{2}{x - 4}\right) + 10 = 0$$
Multiply the equation sides by the denominators:
x and -4 + x
we get:
$$x \left(\left(\left(4 - \frac{7}{x}\right) + \frac{2}{x - 4}\right) + 10\right) = 0$$
$$\frac{14 x^{2} - 61 x + 28}{x - 4} = 0$$
$$\frac{14 x^{2} - 61 x + 28}{x - 4} \left(x - 4\right) = 0 \left(x - 4\right)$$
$$14 x^{2} - 61 x + 28 = 0$$
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
$$a = 14$$
$$b = -61$$
$$c = 28$$
, then
D = b^2 - 4 * a * c =
(-61)^2 - 4 * (14) * (28) = 2153
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{\sqrt{2153}}{28} + \frac{61}{28}$$
$$x_{2} = \frac{61}{28} - \frac{\sqrt{2153}}{28}$$
Sum and product of roots
[src]
sum
______ ______ 61 \/ 2153 61 \/ 2153 -- - -------- + -- + -------- 28 28 28 28
$$\left(\frac{61}{28} - \frac{\sqrt{2153}}{28}\right) + \left(\frac{\sqrt{2153}}{28} + \frac{61}{28}\right)$$
=
61 -- 14
$$\frac{61}{14}$$
product
/ ______\ / ______\ |61 \/ 2153 | |61 \/ 2153 | |-- - --------|*|-- + --------| \28 28 / \28 28 /
$$\left(\frac{61}{28} - \frac{\sqrt{2153}}{28}\right) \left(\frac{\sqrt{2153}}{28} + \frac{61}{28}\right)$$
=
2
$$2$$
2
Rapid solution
[src]
______
61 \/ 2153
x1 = -- - --------
28 28
$$x_{1} = \frac{61}{28} - \frac{\sqrt{2153}}{28}$$
______
61 \/ 2153
x2 = -- + --------
28 28
$$x_{2} = \frac{\sqrt{2153}}{28} + \frac{61}{28}$$
x2 = sqrt(2153)/28 + 61/28