Mister Exam
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How to use it?
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Equation x^2+6=5x
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Identical expressions
- (thirty-six /(nine +x))+(sixteen /(seven +x))= five
- (36 divide by (9 plus x)) plus (16 divide by (7 plus x)) equally 5
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Similar expressions
- (36/(9+x))+(16/(7-x))=5
- (36/(9-x))+(16/(7+x))=5
- (36/(9+x))-(16/(7+x))=5
(36/(9+x))+(16/(7+x))=5 equation
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The solution
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[src]
36 16 ----- + ----- = 5 9 + x 7 + x
$$\frac{36}{x + 9} + \frac{16}{x + 7} = 5$$
Detail solution
Given the equation:
$$\frac{36}{x + 9} + \frac{16}{x + 7} = 5$$
Multiply the equation sides by the denominators:
7 + x and 9 + x
we get:
$$\left(x + 7\right) \left(\frac{36}{x + 9} + \frac{16}{x + 7}\right) = 5 x + 35$$
$$\frac{4 \left(13 x + 99\right)}{x + 9} = 5 x + 35$$
$$\frac{4 \left(13 x + 99\right)}{x + 9} \left(x + 9\right) = \left(x + 9\right) \left(5 x + 35\right)$$
$$52 x + 396 = 5 x^{2} + 80 x + 315$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$52 x + 396 = 5 x^{2} + 80 x + 315$$
to
$$- 5 x^{2} - 28 x + 81 = 0$$
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
$$a = -5$$
$$b = -28$$
$$c = 81$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = - \frac{\sqrt{601}}{5} - \frac{14}{5}$$
$$x_{2} = - \frac{14}{5} + \frac{\sqrt{601}}{5}$$
$$\frac{36}{x + 9} + \frac{16}{x + 7} = 5$$
Multiply the equation sides by the denominators:
7 + x and 9 + x
we get:
$$\left(x + 7\right) \left(\frac{36}{x + 9} + \frac{16}{x + 7}\right) = 5 x + 35$$
$$\frac{4 \left(13 x + 99\right)}{x + 9} = 5 x + 35$$
$$\frac{4 \left(13 x + 99\right)}{x + 9} \left(x + 9\right) = \left(x + 9\right) \left(5 x + 35\right)$$
$$52 x + 396 = 5 x^{2} + 80 x + 315$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$52 x + 396 = 5 x^{2} + 80 x + 315$$
to
$$- 5 x^{2} - 28 x + 81 = 0$$
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
$$a = -5$$
$$b = -28$$
$$c = 81$$
, then
D = b^2 - 4 * a * c =
(-28)^2 - 4 * (-5) * (81) = 2404
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{\sqrt{601}}{5} - \frac{14}{5}$$
$$x_{2} = - \frac{14}{5} + \frac{\sqrt{601}}{5}$$
Sum and product of roots
[src]
sum
_____ _____ 14 \/ 601 14 \/ 601 - -- + ------- + - -- - ------- 5 5 5 5
$$\left(- \frac{\sqrt{601}}{5} - \frac{14}{5}\right) + \left(- \frac{14}{5} + \frac{\sqrt{601}}{5}\right)$$
=
-28/5
$$- \frac{28}{5}$$
product
/ _____\ / _____\ | 14 \/ 601 | | 14 \/ 601 | |- -- + -------|*|- -- - -------| \ 5 5 / \ 5 5 /
$$\left(- \frac{14}{5} + \frac{\sqrt{601}}{5}\right) \left(- \frac{\sqrt{601}}{5} - \frac{14}{5}\right)$$
=
-81/5
$$- \frac{81}{5}$$
-81/5
Rapid solution
[src]
_____
14 \/ 601
x1 = - -- + -------
5 5
$$x_{1} = - \frac{14}{5} + \frac{\sqrt{601}}{5}$$
_____
14 \/ 601
x2 = - -- - -------
5 5
$$x_{2} = - \frac{\sqrt{601}}{5} - \frac{14}{5}$$
x2 = -sqrt(601)/5 - 14/5