Mister Exam
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How to use it?
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Identical expressions
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Similar expressions
- 9^x+26*3^x-27=0
- 9^x-26*3^x+27=0
9^x-26*3^x-27=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$\left(- 26 \cdot 3^{x} + 9^{x}\right) - 27 = 0$$
or
$$\left(- 26 \cdot 3^{x} + 9^{x}\right) - 27 = 0$$
Do replacement
$$v = 3^{x}$$
we get
$$v^{2} - 26 v - 27 = 0$$
or
$$v^{2} - 26 v - 27 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$v_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$v_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -26$$
$$c = -27$$
, then
Because D > 0, then the equation has two roots.
or
$$v_{1} = 27$$
$$v_{2} = -1$$
do backward replacement
$$3^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(3 \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(27 \right)}}{\log{\left(3 \right)}} = 3$$
$$x_{2} = \frac{\log{\left(-1 \right)}}{\log{\left(3 \right)}} = \frac{i \pi}{\log{\left(3 \right)}}$$
$$\left(- 26 \cdot 3^{x} + 9^{x}\right) - 27 = 0$$
or
$$\left(- 26 \cdot 3^{x} + 9^{x}\right) - 27 = 0$$
Do replacement
$$v = 3^{x}$$
we get
$$v^{2} - 26 v - 27 = 0$$
or
$$v^{2} - 26 v - 27 = 0$$
This equation is of the form
a*v^2 + b*v + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$v_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$v_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -26$$
$$c = -27$$
, then
D = b^2 - 4 * a * c =
(-26)^2 - 4 * (1) * (-27) = 784
Because D > 0, then the equation has two roots.
v1 = (-b + sqrt(D)) / (2*a)
v2 = (-b - sqrt(D)) / (2*a)
or
$$v_{1} = 27$$
$$v_{2} = -1$$
do backward replacement
$$3^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(3 \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(27 \right)}}{\log{\left(3 \right)}} = 3$$
$$x_{2} = \frac{\log{\left(-1 \right)}}{\log{\left(3 \right)}} = \frac{i \pi}{\log{\left(3 \right)}}$$
Rapid solution
[src]
x1 = 3
$$x_{1} = 3$$
pi*I
x2 = ------
log(3)
$$x_{2} = \frac{i \pi}{\log{\left(3 \right)}}$$
x2 = i*pi/log(3)
Sum and product of roots
[src]
sum
pi*I
3 + ------
log(3)
$$3 + \frac{i \pi}{\log{\left(3 \right)}}$$
=
pi*I
3 + ------
log(3)
$$3 + \frac{i \pi}{\log{\left(3 \right)}}$$
product
pi*I 3*------ log(3)
$$3 \frac{i \pi}{\log{\left(3 \right)}}$$
=
3*pi*I ------ log(3)
$$\frac{3 i \pi}{\log{\left(3 \right)}}$$
3*pi*i/log(3)