Mister Exam
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How to use it?
- Equation:
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Equation sin(x)-cos(x)=1
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Equation (11x+14)-(5x-8)=25
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Equation sqrt(x^3-4*x^2-10*x+29)=3-x
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Equation cos(x/3)=0
- Express {x} in terms of y where:
- -14*x-16*y=14
- -2*x-15*y=-1
- 4*x+15*y=19
- -16*x-2*y=4
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Identical expressions
- √(x- thirteen)-√ ten -x= two
- √(x minus 13) minus √10 minus x equally 2
- √(x minus thirteen) minus √ ten minus x equally two
- √x-13-√10-x=2
-
Similar expressions
- √(x+13)-√10-x=2
- √(x-13)+√10-x=2
- √(x-13)-√10+x=2
√(x-13)-√10-x=2 equation
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The solution
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[src]
________ ____ \/ x - 13 - \/ 10 - x = 2
$$- x + \left(\sqrt{x - 13} - \sqrt{10}\right) = 2$$
Detail solution
Given the equation
$$- x + \left(\sqrt{x - 13} - \sqrt{10}\right) = 2$$
$$\sqrt{x - 13} = x + 2 + \sqrt{10}$$
We raise the equation sides to 2-th degree
$$x - 13 = \left(x + 2 + \sqrt{10}\right)^{2}$$
$$x - 13 = x^{2} + 4 x + 2 \sqrt{10} x + 4 \sqrt{10} + 14$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} - 2 \sqrt{10} x - 3 x - 27 - 4 \sqrt{10} = 0$$
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
$$a = -1$$
$$b = - 2 \sqrt{10} - 3$$
$$c = -27 - 4 \sqrt{10}$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \sqrt{10} - \frac{3}{2} - \frac{\sqrt{-108 - 16 \sqrt{10} + \left(- 2 \sqrt{10} - 3\right)^{2}}}{2}$$
$$x_{2} = - \sqrt{10} - \frac{3}{2} + \frac{\sqrt{-108 - 16 \sqrt{10} + \left(- 2 \sqrt{10} - 3\right)^{2}}}{2}$$
$$- x + \left(\sqrt{x - 13} - \sqrt{10}\right) = 2$$
$$\sqrt{x - 13} = x + 2 + \sqrt{10}$$
We raise the equation sides to 2-th degree
$$x - 13 = \left(x + 2 + \sqrt{10}\right)^{2}$$
$$x - 13 = x^{2} + 4 x + 2 \sqrt{10} x + 4 \sqrt{10} + 14$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} - 2 \sqrt{10} x - 3 x - 27 - 4 \sqrt{10} = 0$$
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
$$a = -1$$
$$b = - 2 \sqrt{10} - 3$$
$$c = -27 - 4 \sqrt{10}$$
, then
D = b^2 - 4 * a * c =
(-3 - 2*sqrt(10))^2 - 4 * (-1) * (-27 - 4*sqrt(10)) = -108 + (-3 - 2*sqrt(10))^2 - 16*sqrt(10)
Because D<0, then the equation
has no real roots,
but complex roots is exists.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \sqrt{10} - \frac{3}{2} - \frac{\sqrt{-108 - 16 \sqrt{10} + \left(- 2 \sqrt{10} - 3\right)^{2}}}{2}$$
$$x_{2} = - \sqrt{10} - \frac{3}{2} + \frac{\sqrt{-108 - 16 \sqrt{10} + \left(- 2 \sqrt{10} - 3\right)^{2}}}{2}$$
Sum and product of roots
[src]
sum
_______________ _______________
/ ____ / ____
3 ____ I*\/ 59 + 4*\/ 10 3 ____ I*\/ 59 + 4*\/ 10
- - - \/ 10 - -------------------- + - - - \/ 10 + --------------------
2 2 2 2
$$\left(- \sqrt{10} - \frac{3}{2} - \frac{i \sqrt{4 \sqrt{10} + 59}}{2}\right) + \left(- \sqrt{10} - \frac{3}{2} + \frac{i \sqrt{4 \sqrt{10} + 59}}{2}\right)$$
=
____ -3 - 2*\/ 10
$$- 2 \sqrt{10} - 3$$
product
/ _______________\ / _______________\ | / ____ | | / ____ | | 3 ____ I*\/ 59 + 4*\/ 10 | | 3 ____ I*\/ 59 + 4*\/ 10 | |- - - \/ 10 - --------------------|*|- - - \/ 10 + --------------------| \ 2 2 / \ 2 2 /
$$\left(- \sqrt{10} - \frac{3}{2} - \frac{i \sqrt{4 \sqrt{10} + 59}}{2}\right) \left(- \sqrt{10} - \frac{3}{2} + \frac{i \sqrt{4 \sqrt{10} + 59}}{2}\right)$$
=
____ 27 + 4*\/ 10
$$4 \sqrt{10} + 27$$
27 + 4*sqrt(10)
Rapid solution
[src]
_______________
/ ____
3 ____ I*\/ 59 + 4*\/ 10
x1 = - - - \/ 10 - --------------------
2 2
$$x_{1} = - \sqrt{10} - \frac{3}{2} - \frac{i \sqrt{4 \sqrt{10} + 59}}{2}$$
_______________
/ ____
3 ____ I*\/ 59 + 4*\/ 10
x2 = - - - \/ 10 + --------------------
2 2
$$x_{2} = - \sqrt{10} - \frac{3}{2} + \frac{i \sqrt{4 \sqrt{10} + 59}}{2}$$
x2 = -sqrt(10) - 3/2 + i*sqrt(4*sqrt(10) + 59)/2
Numerical answer
[src]
x1 = -4.66227766016838 - 4.23228988375895*i
x2 = -4.66227766016838 + 4.23228988375895*i
x2 = -4.66227766016838 + 4.23228988375895*i