Mister Exam
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Identical expressions
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Similar expressions
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(3x-2)(3x-1)3x=504 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
(3*x - 2)*(3*x - 1)*3*x = 504
$$x 3 \left(3 x - 2\right) \left(3 x - 1\right) = 504$$
Detail solution
Given the equation:
$$x 3 \left(3 x - 2\right) \left(3 x - 1\right) = 504$$
transform:
Take common factor from the equation
$$3 \left(x - 3\right) \left(9 x^{2} + 18 x + 56\right) = 0$$
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
$$3 x - 9 = 0$$
$$9 x^{2} + 18 x + 56 = 0$$
solve the resulting equation:
1.
$$3 x - 9 = 0$$
Move free summands (without x)
from left part to right part, we given:
$$3 x = 9$$
Divide both parts of the equation by 3
We get the answer: x1 = 3
2.
$$9 x^{2} + 18 x + 56 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 9$$
$$b = 18$$
$$c = 56$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{2} = -1 + \frac{\sqrt{47} i}{3}$$
$$x_{3} = -1 - \frac{\sqrt{47} i}{3}$$
The final answer:
$$x_{1} = 3$$
$$x_{2} = -1 + \frac{\sqrt{47} i}{3}$$
$$x_{3} = -1 - \frac{\sqrt{47} i}{3}$$
$$x 3 \left(3 x - 2\right) \left(3 x - 1\right) = 504$$
transform:
Take common factor from the equation
$$3 \left(x - 3\right) \left(9 x^{2} + 18 x + 56\right) = 0$$
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
$$3 x - 9 = 0$$
$$9 x^{2} + 18 x + 56 = 0$$
solve the resulting equation:
1.
$$3 x - 9 = 0$$
Move free summands (without x)
from left part to right part, we given:
$$3 x = 9$$
Divide both parts of the equation by 3
x = 9 / (3)
We get the answer: x1 = 3
2.
$$9 x^{2} + 18 x + 56 = 0$$
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:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 9$$
$$b = 18$$
$$c = 56$$
, then
D = b^2 - 4 * a * c =
(18)^2 - 4 * (9) * (56) = -1692
Because D<0, then the equation
has no real roots,
but complex roots is exists.
x2 = (-b + sqrt(D)) / (2*a)
x3 = (-b - sqrt(D)) / (2*a)
or
$$x_{2} = -1 + \frac{\sqrt{47} i}{3}$$
$$x_{3} = -1 - \frac{\sqrt{47} i}{3}$$
The final answer:
$$x_{1} = 3$$
$$x_{2} = -1 + \frac{\sqrt{47} i}{3}$$
$$x_{3} = -1 - \frac{\sqrt{47} i}{3}$$
Sum and product of roots
[src]
sum
____ ____
I*\/ 47 I*\/ 47
3 + -1 - -------- + -1 + --------
3 3
$$\left(3 + \left(-1 - \frac{\sqrt{47} i}{3}\right)\right) + \left(-1 + \frac{\sqrt{47} i}{3}\right)$$
=
1
$$1$$
product
/ ____\ / ____\ | I*\/ 47 | | I*\/ 47 | 3*|-1 - --------|*|-1 + --------| \ 3 / \ 3 /
$$3 \left(-1 - \frac{\sqrt{47} i}{3}\right) \left(-1 + \frac{\sqrt{47} i}{3}\right)$$
=
56/3
$$\frac{56}{3}$$
56/3
Rapid solution
[src]
x1 = 3
$$x_{1} = 3$$
____
I*\/ 47
x2 = -1 - --------
3
$$x_{2} = -1 - \frac{\sqrt{47} i}{3}$$
____
I*\/ 47
x3 = -1 + --------
3
$$x_{3} = -1 + \frac{\sqrt{47} i}{3}$$
x3 = -1 + sqrt(47)*i/3
Numerical answer
[src]
x1 = 3.0
x2 = -1.0 + 2.28521820013368*i
x3 = -1.0 - 2.28521820013368*i
x3 = -1.0 - 2.28521820013368*i