Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation sin(x)=√3/2
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Equation sin(x)=-3/2
-
Equation (x-1)(x+2)-x(x-3)+5=x+4
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Equation (3*x+4)/(x^2-16)=x^2/(x^2-16)
- Express {x} in terms of y where:
- 20*x-16*y=6
- -5*x-19*y=18
- 17*x+11*y=10
- -9*x-15*y=-17
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Identical expressions
- x^ two + three *(one -sqrt(six))*x+ eighteen / five *(five + two *sqrt(six))= zero
- x squared plus 3 multiply by (1 minus square root of (6)) multiply by x plus 18 divide by 5 multiply by (5 plus 2 multiply by square root of (6)) equally 0
- x to the power of two plus three multiply by (one minus square root of (six)) multiply by x plus eighteen divide by five multiply by (five plus two multiply by square root of (six)) equally zero
- x^2+3*(1-√(6))*x+18/5*(5+2*√(6))=0
- x2+3*(1-sqrt(6))*x+18/5*(5+2*sqrt(6))=0
- x2+3*1-sqrt6*x+18/5*5+2*sqrt6=0
- x²+3*(1-sqrt(6))*x+18/5*(5+2*sqrt(6))=0
- x to the power of 2+3*(1-sqrt(6))*x+18/5*(5+2*sqrt(6))=0
- x^2+3(1-sqrt(6))x+18/5(5+2sqrt(6))=0
- x2+3(1-sqrt(6))x+18/5(5+2sqrt(6))=0
- x2+31-sqrt6x+18/55+2sqrt6=0
- x^2+31-sqrt6x+18/55+2sqrt6=0
- x^2+3*(1-sqrt(6))*x+18/5*(5+2*sqrt(6))=O
- x^2+3*(1-sqrt(6))*x+18 divide by 5*(5+2*sqrt(6))=0
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Similar expressions
- x^2-3*(1-sqrt(6))*x+18/5*(5+2*sqrt(6))=0
- x^2+3*(1-sqrt(6))*x+18/5*(5-2*sqrt(6))=0
- x^2+3*(1-sqrt(6))*x-18/5*(5+2*sqrt(6))=0
- x^2+3*(1+sqrt(6))*x+18/5*(5+2*sqrt(6))=0
x^2+3*(1-sqrt(6))*x+18/5*(5+2*sqrt(6))=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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/ ___\
2 / ___\ 18*\5 + 2*\/ 6 /
x + 3*\1 - \/ 6 /*x + ---------------- = 0
5
$$\left(x^{2} + x 3 \left(1 - \sqrt{6}\right)\right) + \frac{18 \left(2 \sqrt{6} + 5\right)}{5} = 0$$
Detail solution
Expand the expression in the equation
$$\left(x^{2} + x 3 \left(1 - \sqrt{6}\right)\right) + \frac{18 \left(2 \sqrt{6} + 5\right)}{5} = 0$$
We get the quadratic equation
$$x^{2} - 3 \sqrt{6} x + 3 x + \frac{36 \sqrt{6}}{5} + 18 = 0$$
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
$$a = 1$$
$$b = 3 - 3 \sqrt{6}$$
$$c = \frac{36 \sqrt{6}}{5} + 18$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \frac{3}{2} + \frac{3 \sqrt{6}}{2} + \frac{\sqrt{-72 - \frac{144 \sqrt{6}}{5} + \left(3 - 3 \sqrt{6}\right)^{2}}}{2}$$
$$x_{2} = - \frac{3}{2} + \frac{3 \sqrt{6}}{2} - \frac{\sqrt{-72 - \frac{144 \sqrt{6}}{5} + \left(3 - 3 \sqrt{6}\right)^{2}}}{2}$$
$$\left(x^{2} + x 3 \left(1 - \sqrt{6}\right)\right) + \frac{18 \left(2 \sqrt{6} + 5\right)}{5} = 0$$
We get the quadratic equation
$$x^{2} - 3 \sqrt{6} x + 3 x + \frac{36 \sqrt{6}}{5} + 18 = 0$$
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
$$a = 1$$
$$b = 3 - 3 \sqrt{6}$$
$$c = \frac{36 \sqrt{6}}{5} + 18$$
, then
D = b^2 - 4 * a * c =
(3 - 3*sqrt(6))^2 - 4 * (1) * (18 + 36*sqrt(6)/5) = -72 + (3 - 3*sqrt(6))^2 - 144*sqrt(6)/5
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
$$x_{1} = - \frac{3}{2} + \frac{3 \sqrt{6}}{2} + \frac{\sqrt{-72 - \frac{144 \sqrt{6}}{5} + \left(3 - 3 \sqrt{6}\right)^{2}}}{2}$$
$$x_{2} = - \frac{3}{2} + \frac{3 \sqrt{6}}{2} - \frac{\sqrt{-72 - \frac{144 \sqrt{6}}{5} + \left(3 - 3 \sqrt{6}\right)^{2}}}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 3 - 3 \sqrt{6}$$
$$q = \frac{c}{a}$$
$$q = \frac{18 \left(2 \sqrt{6} + 5\right)}{5}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -3 + 3 \sqrt{6}$$
$$x_{1} x_{2} = \frac{18 \left(2 \sqrt{6} + 5\right)}{5}$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 3 - 3 \sqrt{6}$$
$$q = \frac{c}{a}$$
$$q = \frac{18 \left(2 \sqrt{6} + 5\right)}{5}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -3 + 3 \sqrt{6}$$
$$x_{1} x_{2} = \frac{18 \left(2 \sqrt{6} + 5\right)}{5}$$
Sum and product of roots
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sum
________________ ________________
___ / ___ ___ / ___
3 3*\/ 6 3*I*\/ 25 + 130*\/ 6 3 3*\/ 6 3*I*\/ 25 + 130*\/ 6
- - + ------- - ----------------------- + - - + ------- + -----------------------
2 2 10 2 2 10
$$\left(- \frac{3}{2} + \frac{3 \sqrt{6}}{2} - \frac{3 i \sqrt{25 + 130 \sqrt{6}}}{10}\right) + \left(- \frac{3}{2} + \frac{3 \sqrt{6}}{2} + \frac{3 i \sqrt{25 + 130 \sqrt{6}}}{10}\right)$$
=
___ -3 + 3*\/ 6
$$-3 + 3 \sqrt{6}$$
product
/ ________________\ / ________________\ | ___ / ___ | | ___ / ___ | | 3 3*\/ 6 3*I*\/ 25 + 130*\/ 6 | | 3 3*\/ 6 3*I*\/ 25 + 130*\/ 6 | |- - + ------- - -----------------------|*|- - + ------- + -----------------------| \ 2 2 10 / \ 2 2 10 /
$$\left(- \frac{3}{2} + \frac{3 \sqrt{6}}{2} - \frac{3 i \sqrt{25 + 130 \sqrt{6}}}{10}\right) \left(- \frac{3}{2} + \frac{3 \sqrt{6}}{2} + \frac{3 i \sqrt{25 + 130 \sqrt{6}}}{10}\right)$$
=
___
36*\/ 6
18 + --------
5
$$\frac{36 \sqrt{6}}{5} + 18$$
18 + 36*sqrt(6)/5
Rapid solution
[src]
________________
___ / ___
3 3*\/ 6 3*I*\/ 25 + 130*\/ 6
x1 = - - + ------- - -----------------------
2 2 10
$$x_{1} = - \frac{3}{2} + \frac{3 \sqrt{6}}{2} - \frac{3 i \sqrt{25 + 130 \sqrt{6}}}{10}$$
________________
___ / ___
3 3*\/ 6 3*I*\/ 25 + 130*\/ 6
x2 = - - + ------- + -----------------------
2 2 10
$$x_{2} = - \frac{3}{2} + \frac{3 \sqrt{6}}{2} + \frac{3 i \sqrt{25 + 130 \sqrt{6}}}{10}$$
x2 = -3/2 + 3*sqrt(6)/2 + 3*i*sqrt(25 + 130*sqrt(6))/10
Numerical answer
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x1 = 2.17423461417477 + 5.55958901273855*i
x2 = 2.17423461417477 - 5.55958901273855*i
x2 = 2.17423461417477 - 5.55958901273855*i