Mister Exam
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How to use it?
- Equation:
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Equation x^4+4=0
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Equation sin(x^2)=0
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Equation -x^2+5*x-6=0
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Equation (x-119)/(x+7)=-5
- Express {x} in terms of y where:
- 16*x+7*y=8
- 15*x-3*y=19
- -4*x-18*y=-14
- 18*x-20*y=-7
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Identical expressions
- log two *(x^ two - five)*log3^ two *(seven -x)+3log two *(x^2- five)-2log3^2*(seven -x)- six = zero
- logarithm of 2 multiply by (x squared minus 5) multiply by logarithm of 3 squared multiply by (7 minus x) plus 3 logarithm of 2 multiply by (x squared minus 5) minus 2 logarithm of 3 squared multiply by (7 minus x) minus 6 equally 0
- logarithm of two multiply by (x to the power of two minus five) multiply by logarithm of 3 to the power of two multiply by (seven minus x) plus 3 logarithm of two multiply by (x squared minus five) minus 2 logarithm of 3 squared multiply by (seven minus x) minus six equally zero
- log2*(x2-5)*log32*(7-x)+3log2*(x2-5)-2log32*(7-x)-6=0
- log2*x2-5*log32*7-x+3log2*x2-5-2log32*7-x-6=0
- log2*(x²-5)*log3²*(7-x)+3log2*(x²-5)-2log3²*(7-x)-6=0
- log2*(x to the power of 2-5)*log3 to the power of 2*(7-x)+3log2*(x to the power of 2-5)-2log3 to the power of 2*(7-x)-6=0
- log2(x^2-5)log3^2(7-x)+3log2(x^2-5)-2log3^2(7-x)-6=0
- log2(x2-5)log32(7-x)+3log2(x2-5)-2log32(7-x)-6=0
- log2x2-5log327-x+3log2x2-5-2log327-x-6=0
- log2x^2-5log3^27-x+3log2x^2-5-2log3^27-x-6=0
- log2*(x^2-5)*log3^2*(7-x)+3log2*(x^2-5)-2log3^2*(7-x)-6=O
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Similar expressions
- log2*(x^2-5)*log3^2*(7+x)+3log2*(x^2-5)-2log3^2*(7-x)-6=0
- log2*(x^2-5)*log3^2*(7-x)+3log2*(x^2-5)+2log3^2*(7-x)-6=0
- log2*(x^2-5)*log3^2*(7-x)+3log2*(x^2-5)-2log3^2*(7-x)+6=0
- log2*(x^2+5)*log3^2*(7-x)+3log2*(x^2-5)-2log3^2*(7-x)-6=0
- log2*(x^2-5)*log3^2*(7-x)+3log2*(x^2+5)-2log3^2*(7-x)-6=0
- log2*(x^2-5)*log3^2*(7-x)-3log2*(x^2-5)-2log3^2*(7-x)-6=0
- log2*(x^2-5)*log3^2*(7-x)+3log2*(x^2-5)-2log3^2*(7+x)-6=0
log2*(x^2-5)*log3^2*(7-x)+3log2*(x^2-5)-2log3^2*(7-x)-6=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
/ 2 \ 2 / 2 \ 2 log(2)*\x - 5/*log (3)*(7 - x) + 3*log(2)*\x - 5/ - 2*log (3)*(7 - x) - 6 = 0
$$\left(- \left(7 - x\right) 2 \log{\left(3 \right)}^{2} + \left(\left(x^{2} - 5\right) \log{\left(2 \right)} \log{\left(3 \right)}^{2} \left(7 - x\right) + \left(x^{2} - 5\right) 3 \log{\left(2 \right)}\right)\right) - 6 = 0$$
Detail solution
Given the equation:
$$\left(- \left(7 - x\right) 2 \log{\left(3 \right)}^{2} + \left(\left(x^{2} - 5\right) \log{\left(2 \right)} \log{\left(3 \right)}^{2} \left(7 - x\right) + \left(x^{2} - 5\right) 3 \log{\left(2 \right)}\right)\right) - 6 = 0$$
transform:
Take common factor from the equation
$$- \left(x \log{\left(3 \right)}^{2} - 7 \log{\left(3 \right)}^{2} - 3\right) \left(x^{2} \log{\left(2 \right)} - 5 \log{\left(2 \right)} - 2\right) = 0$$
Because the right side of the equation is zero, then the solution of the equation is exists if at least one of the multipliers in the left side of the equation equal to zero.
We get the equations
$$- x^{2} \log{\left(2 \right)} + 2 + 5 \log{\left(2 \right)} = 0$$
$$x \log{\left(3 \right)}^{2} - 7 \log{\left(3 \right)}^{2} - 3 = 0$$
solve the resulting equation:
1.
$$- x^{2} \log{\left(2 \right)} + 2 + 5 \log{\left(2 \right)} = 0$$
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
$$a = - \log{\left(2 \right)}$$
$$b = 0$$
$$c = 2 + 5 \log{\left(2 \right)}$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = - \frac{\sqrt{2 + 5 \log{\left(2 \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
$$x_{2} = \frac{\sqrt{2 + 5 \log{\left(2 \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
2.
$$x \log{\left(3 \right)}^{2} - 7 \log{\left(3 \right)}^{2} - 3 = 0$$
Expand brackets in the left part
Move free summands (without x)
from left part to right part, we given:
$$x \log{\left(3 \right)}^{2} - 7 \log{\left(3 \right)}^{2} = 3$$
Divide both parts of the equation by (-7*log(3)^2 + x*log(3)^2)/x
We get the answer: x3 = 7 + 3/log(3)^2
The final answer:
$$x_{1} = - \frac{\sqrt{2 + 5 \log{\left(2 \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
$$x_{2} = \frac{\sqrt{2 + 5 \log{\left(2 \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
$$x_{3} = \frac{3}{\log{\left(3 \right)}^{2}} + 7$$
$$\left(- \left(7 - x\right) 2 \log{\left(3 \right)}^{2} + \left(\left(x^{2} - 5\right) \log{\left(2 \right)} \log{\left(3 \right)}^{2} \left(7 - x\right) + \left(x^{2} - 5\right) 3 \log{\left(2 \right)}\right)\right) - 6 = 0$$
transform:
Take common factor from the equation
$$- \left(x \log{\left(3 \right)}^{2} - 7 \log{\left(3 \right)}^{2} - 3\right) \left(x^{2} \log{\left(2 \right)} - 5 \log{\left(2 \right)} - 2\right) = 0$$
Because the right side of the equation is zero, then the solution of the equation is exists if at least one of the multipliers in the left side of the equation equal to zero.
We get the equations
$$- x^{2} \log{\left(2 \right)} + 2 + 5 \log{\left(2 \right)} = 0$$
$$x \log{\left(3 \right)}^{2} - 7 \log{\left(3 \right)}^{2} - 3 = 0$$
solve the resulting equation:
1.
$$- x^{2} \log{\left(2 \right)} + 2 + 5 \log{\left(2 \right)} = 0$$
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
$$a = - \log{\left(2 \right)}$$
$$b = 0$$
$$c = 2 + 5 \log{\left(2 \right)}$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (-log(2)) * (2 + 5*log(2)) = 4*(2 + 5*log(2))*log(2)
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{\sqrt{2 + 5 \log{\left(2 \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
$$x_{2} = \frac{\sqrt{2 + 5 \log{\left(2 \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
2.
$$x \log{\left(3 \right)}^{2} - 7 \log{\left(3 \right)}^{2} - 3 = 0$$
Expand brackets in the left part
-3 - 7*log3^2 + x*log3^2 = 0
Move free summands (without x)
from left part to right part, we given:
$$x \log{\left(3 \right)}^{2} - 7 \log{\left(3 \right)}^{2} = 3$$
Divide both parts of the equation by (-7*log(3)^2 + x*log(3)^2)/x
x = 3 / ((-7*log(3)^2 + x*log(3)^2)/x)
We get the answer: x3 = 7 + 3/log(3)^2
The final answer:
$$x_{1} = - \frac{\sqrt{2 + 5 \log{\left(2 \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
$$x_{2} = \frac{\sqrt{2 + 5 \log{\left(2 \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
$$x_{3} = \frac{3}{\log{\left(3 \right)}^{2}} + 7$$
Rapid solution
[src]
______________
-\/ 2 + 5*log(2)
x1 = ------------------
________
\/ log(2)
$$x_{1} = - \frac{\sqrt{2 + 5 \log{\left(2 \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
______________
\/ 2 + 5*log(2)
x2 = ----------------
________
\/ log(2)
$$x_{2} = \frac{\sqrt{2 + 5 \log{\left(2 \right)}}}{\sqrt{\log{\left(2 \right)}}}$$
2
3 + 7*log (3)
x3 = -------------
2
log (3)
$$x_{3} = \frac{3 + 7 \log{\left(3 \right)}^{2}}{\log{\left(3 \right)}^{2}}$$
x3 = (3 + 7*log(3)^2)/log(3)^2
Sum and product of roots
[src]
sum
______________ ______________ 2
\/ 2 + 5*log(2) \/ 2 + 5*log(2) 3 + 7*log (3)
- ---------------- + ---------------- + -------------
________ ________ 2
\/ log(2) \/ log(2) log (3)
$$\left(- \frac{\sqrt{2 + 5 \log{\left(2 \right)}}}{\sqrt{\log{\left(2 \right)}}} + \frac{\sqrt{2 + 5 \log{\left(2 \right)}}}{\sqrt{\log{\left(2 \right)}}}\right) + \frac{3 + 7 \log{\left(3 \right)}^{2}}{\log{\left(3 \right)}^{2}}$$
=
2
3 + 7*log (3)
-------------
2
log (3)
$$\frac{3 + 7 \log{\left(3 \right)}^{2}}{\log{\left(3 \right)}^{2}}$$
product
______________ ______________ 2
-\/ 2 + 5*log(2) \/ 2 + 5*log(2) 3 + 7*log (3)
------------------*----------------*-------------
________ ________ 2
\/ log(2) \/ log(2) log (3)
$$- \frac{\sqrt{2 + 5 \log{\left(2 \right)}}}{\sqrt{\log{\left(2 \right)}}} \frac{\sqrt{2 + 5 \log{\left(2 \right)}}}{\sqrt{\log{\left(2 \right)}}} \frac{3 + 7 \log{\left(3 \right)}^{2}}{\log{\left(3 \right)}^{2}}$$
=
/ 2 \
-(2 + log(32))*\3 + 7*log (3)/
-------------------------------
2
log(2)*log (3)
$$- \frac{\left(2 + \log{\left(32 \right)}\right) \left(3 + 7 \log{\left(3 \right)}^{2}\right)}{\log{\left(2 \right)} \log{\left(3 \right)}^{2}}$$
-(2 + log(32))*(3 + 7*log(3)^2)/(log(2)*log(3)^2)
Numerical answer
[src]
x1 = 9.48560634907067
x2 = -2.80809367396779
x3 = 2.80809367396779
x3 = 2.80809367396779