Mister Exam
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(2x-1)(x+1)-x=(x-3)-10 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
(2*x - 1)*(x + 1) - x = x - 3 - 10
$$- x + \left(x + 1\right) \left(2 x - 1\right) = \left(x - 3\right) - 10$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$- x + \left(x + 1\right) \left(2 x - 1\right) = \left(x - 3\right) - 10$$
to
$$\left(- x + \left(x + 1\right) \left(2 x - 1\right)\right) + \left(\left(3 - x\right) + 10\right) = 0$$
Expand the expression in the equation
$$\left(- x + \left(x + 1\right) \left(2 x - 1\right)\right) + \left(\left(3 - x\right) + 10\right) = 0$$
We get the quadratic equation
$$2 x^{2} - x + 12 = 0$$
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
$$a = 2$$
$$b = -1$$
$$c = 12$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{1}{4} + \frac{\sqrt{95} i}{4}$$
$$x_{2} = \frac{1}{4} - \frac{\sqrt{95} i}{4}$$
left part with negative sign.
The equation is transformed from
$$- x + \left(x + 1\right) \left(2 x - 1\right) = \left(x - 3\right) - 10$$
to
$$\left(- x + \left(x + 1\right) \left(2 x - 1\right)\right) + \left(\left(3 - x\right) + 10\right) = 0$$
Expand the expression in the equation
$$\left(- x + \left(x + 1\right) \left(2 x - 1\right)\right) + \left(\left(3 - x\right) + 10\right) = 0$$
We get the quadratic equation
$$2 x^{2} - x + 12 = 0$$
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
$$a = 2$$
$$b = -1$$
$$c = 12$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (2) * (12) = -95
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
$$x_{1} = \frac{1}{4} + \frac{\sqrt{95} i}{4}$$
$$x_{2} = \frac{1}{4} - \frac{\sqrt{95} i}{4}$$
Rapid solution
[src]
____
1 I*\/ 95
x1 = - - --------
4 4
$$x_{1} = \frac{1}{4} - \frac{\sqrt{95} i}{4}$$
____
1 I*\/ 95
x2 = - + --------
4 4
$$x_{2} = \frac{1}{4} + \frac{\sqrt{95} i}{4}$$
x2 = 1/4 + sqrt(95)*i/4
Sum and product of roots
[src]
sum
____ ____ 1 I*\/ 95 1 I*\/ 95 - - -------- + - + -------- 4 4 4 4
$$\left(\frac{1}{4} - \frac{\sqrt{95} i}{4}\right) + \left(\frac{1}{4} + \frac{\sqrt{95} i}{4}\right)$$
=
1/2
$$\frac{1}{2}$$
product
/ ____\ / ____\ |1 I*\/ 95 | |1 I*\/ 95 | |- - --------|*|- + --------| \4 4 / \4 4 /
$$\left(\frac{1}{4} - \frac{\sqrt{95} i}{4}\right) \left(\frac{1}{4} + \frac{\sqrt{95} i}{4}\right)$$
=
6
$$6$$
6
Numerical answer
[src]
x1 = 0.25 - 2.43669858620224*i
x2 = 0.25 + 2.43669858620224*i
x2 = 0.25 + 2.43669858620224*i