Mister Exam
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Similar expressions
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x-sqrt(x)+12=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\left(- \sqrt{x} + x\right) + 12 = 0$$
$$- \sqrt{x} = - x - 12$$
We raise the equation sides to 2-th degree
$$x = \left(- x - 12\right)^{2}$$
$$x = x^{2} + 24 x + 144$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} - 23 x - 144 = 0$$
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
$$a = -1$$
$$b = -23$$
$$c = -144$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \frac{23}{2} - \frac{\sqrt{47} i}{2}$$
$$x_{2} = - \frac{23}{2} + \frac{\sqrt{47} i}{2}$$
$$\left(- \sqrt{x} + x\right) + 12 = 0$$
$$- \sqrt{x} = - x - 12$$
We raise the equation sides to 2-th degree
$$x = \left(- x - 12\right)^{2}$$
$$x = x^{2} + 24 x + 144$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} - 23 x - 144 = 0$$
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
$$a = -1$$
$$b = -23$$
$$c = -144$$
, then
D = b^2 - 4 * a * c =
(-23)^2 - 4 * (-1) * (-144) = -47
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
$$x_{1} = - \frac{23}{2} - \frac{\sqrt{47} i}{2}$$
$$x_{2} = - \frac{23}{2} + \frac{\sqrt{47} i}{2}$$
Rapid solution
[src]
____
23 I*\/ 47
x1 = - -- - --------
2 2
$$x_{1} = - \frac{23}{2} - \frac{\sqrt{47} i}{2}$$
____
23 I*\/ 47
x2 = - -- + --------
2 2
$$x_{2} = - \frac{23}{2} + \frac{\sqrt{47} i}{2}$$
x2 = -23/2 + sqrt(47)*i/2
Sum and product of roots
[src]
sum
____ ____ 23 I*\/ 47 23 I*\/ 47 - -- - -------- + - -- + -------- 2 2 2 2
$$\left(- \frac{23}{2} - \frac{\sqrt{47} i}{2}\right) + \left(- \frac{23}{2} + \frac{\sqrt{47} i}{2}\right)$$
=
-23
$$-23$$
product
/ ____\ / ____\ | 23 I*\/ 47 | | 23 I*\/ 47 | |- -- - --------|*|- -- + --------| \ 2 2 / \ 2 2 /
$$\left(- \frac{23}{2} - \frac{\sqrt{47} i}{2}\right) \left(- \frac{23}{2} + \frac{\sqrt{47} i}{2}\right)$$
=
144
$$144$$
144
Numerical answer
[src]
x1 = -11.5 - 3.42782730020052*i
x2 = -11.5 + 3.42782730020052*i
x2 = -11.5 + 3.42782730020052*i