Mister Exam
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How to use it?
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Similar expressions
- ((1)/(x-4)^2)-((7)/(x+4))+(10)=0
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((1)/(x-4)^2)-((7)/(x-4))+(10)=0 equation
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The solution
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[src]
1 7
-------- - ----- + 10 = 0
2 x - 4
(x - 4)
$$\left(\frac{1}{\left(x - 4\right)^{2}} - \frac{7}{x - 4}\right) + 10 = 0$$
Detail solution
Given the equation:
$$\left(\frac{1}{\left(x - 4\right)^{2}} - \frac{7}{x - 4}\right) + 10 = 0$$
Multiply the equation sides by the denominators:
(-4 + x)^2
we get:
$$\left(x - 4\right)^{2} \left(\left(\frac{1}{\left(x - 4\right)^{2}} - \frac{7}{x - 4}\right) + 10\right) = 0$$
$$10 x^{2} - 87 x + 189 = 0$$
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
$$a = 10$$
$$b = -87$$
$$c = 189$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{9}{2}$$
$$x_{2} = \frac{21}{5}$$
$$\left(\frac{1}{\left(x - 4\right)^{2}} - \frac{7}{x - 4}\right) + 10 = 0$$
Multiply the equation sides by the denominators:
(-4 + x)^2
we get:
$$\left(x - 4\right)^{2} \left(\left(\frac{1}{\left(x - 4\right)^{2}} - \frac{7}{x - 4}\right) + 10\right) = 0$$
$$10 x^{2} - 87 x + 189 = 0$$
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
$$a = 10$$
$$b = -87$$
$$c = 189$$
, then
D = b^2 - 4 * a * c =
(-87)^2 - 4 * (10) * (189) = 9
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{9}{2}$$
$$x_{2} = \frac{21}{5}$$
Sum and product of roots
[src]
sum
21/5 + 9/2
$$\frac{21}{5} + \frac{9}{2}$$
=
87 -- 10
$$\frac{87}{10}$$
product
21*9 ---- 5*2
$$\frac{9 \cdot 21}{2 \cdot 5}$$
=
189 --- 10
$$\frac{189}{10}$$
189/10