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Similar expressions
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(x-3)^4-(x-3)^2-10=0 equation
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The solution
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[src]
4 2 (x - 3) - (x - 3) - 10 = 0
$$\left(\left(x - 3\right)^{4} - \left(x - 3\right)^{2}\right) - 10 = 0$$
Detail solution
Given the equation:
$$\left(\left(x - 3\right)^{4} - \left(x - 3\right)^{2}\right) - 10 = 0$$
Do replacement
$$v = \left(x - 3\right)^{2}$$
then the equation will be the:
$$v^{2} - v - 10 = 0$$
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
$$a = 1$$
$$b = -1$$
$$c = -10$$
, then
Because D > 0, then the equation has two roots.
or
$$v_{1} = \frac{1}{2} + \frac{\sqrt{41}}{2}$$
$$v_{2} = \frac{1}{2} - \frac{\sqrt{41}}{2}$$
The final answer:
Because
$$v = \left(x - 3\right)^{2}$$
then
$$x_{1} = \sqrt{v_{1}} + 3$$
$$x_{2} = 3 - \sqrt{v_{1}}$$
$$x_{3} = \sqrt{v_{2}} + 3$$
$$x_{4} = 3 - \sqrt{v_{2}}$$
then:
$$x_{1} = $$
$$\frac{\left(\frac{1}{2} + \frac{\sqrt{41}}{2}\right)^{\frac{1}{2}}}{1} + \frac{3}{1} = \sqrt{\frac{1}{2} + \frac{\sqrt{41}}{2}} + 3$$
$$x_{2} = $$
$$\frac{\left(-1\right) \left(\frac{1}{2} + \frac{\sqrt{41}}{2}\right)^{\frac{1}{2}}}{1} + \frac{3}{1} = 3 - \sqrt{\frac{1}{2} + \frac{\sqrt{41}}{2}}$$
$$x_{3} = $$
$$\frac{3}{1} + \frac{\left(\frac{1}{2} - \frac{\sqrt{41}}{2}\right)^{\frac{1}{2}}}{1} = 3 + \sqrt{\frac{1}{2} - \frac{\sqrt{41}}{2}}$$
$$x_{4} = $$
$$\frac{3}{1} + \frac{\left(-1\right) \left(\frac{1}{2} - \frac{\sqrt{41}}{2}\right)^{\frac{1}{2}}}{1} = 3 - \sqrt{\frac{1}{2} - \frac{\sqrt{41}}{2}}$$
$$\left(\left(x - 3\right)^{4} - \left(x - 3\right)^{2}\right) - 10 = 0$$
Do replacement
$$v = \left(x - 3\right)^{2}$$
then the equation will be the:
$$v^{2} - v - 10 = 0$$
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
$$a = 1$$
$$b = -1$$
$$c = -10$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1) * (-10) = 41
Because D > 0, then the equation has two roots.
v1 = (-b + sqrt(D)) / (2*a)
v2 = (-b - sqrt(D)) / (2*a)
or
$$v_{1} = \frac{1}{2} + \frac{\sqrt{41}}{2}$$
$$v_{2} = \frac{1}{2} - \frac{\sqrt{41}}{2}$$
The final answer:
Because
$$v = \left(x - 3\right)^{2}$$
then
$$x_{1} = \sqrt{v_{1}} + 3$$
$$x_{2} = 3 - \sqrt{v_{1}}$$
$$x_{3} = \sqrt{v_{2}} + 3$$
$$x_{4} = 3 - \sqrt{v_{2}}$$
then:
$$x_{1} = $$
$$\frac{\left(\frac{1}{2} + \frac{\sqrt{41}}{2}\right)^{\frac{1}{2}}}{1} + \frac{3}{1} = \sqrt{\frac{1}{2} + \frac{\sqrt{41}}{2}} + 3$$
$$x_{2} = $$
$$\frac{\left(-1\right) \left(\frac{1}{2} + \frac{\sqrt{41}}{2}\right)^{\frac{1}{2}}}{1} + \frac{3}{1} = 3 - \sqrt{\frac{1}{2} + \frac{\sqrt{41}}{2}}$$
$$x_{3} = $$
$$\frac{3}{1} + \frac{\left(\frac{1}{2} - \frac{\sqrt{41}}{2}\right)^{\frac{1}{2}}}{1} = 3 + \sqrt{\frac{1}{2} - \frac{\sqrt{41}}{2}}$$
$$x_{4} = $$
$$\frac{3}{1} + \frac{\left(-1\right) \left(\frac{1}{2} - \frac{\sqrt{41}}{2}\right)^{\frac{1}{2}}}{1} = 3 - \sqrt{\frac{1}{2} - \frac{\sqrt{41}}{2}}$$
Rapid solution
[src]
____________
___ / ____
\/ 2 *\/ 1 + \/ 41
x1 = 3 - ---------------------
2
$$x_{1} = - \frac{\sqrt{2} \sqrt{1 + \sqrt{41}}}{2} + 3$$
____________
___ / ____
\/ 2 *\/ 1 + \/ 41
x2 = 3 + ---------------------
2
$$x_{2} = \frac{\sqrt{2} \sqrt{1 + \sqrt{41}}}{2} + 3$$
_____________
___ / ____
I*\/ 2 *\/ -1 + \/ 41
x3 = 3 - ------------------------
2
$$x_{3} = 3 - \frac{\sqrt{2} i \sqrt{-1 + \sqrt{41}}}{2}$$
_____________
___ / ____
I*\/ 2 *\/ -1 + \/ 41
x4 = 3 + ------------------------
2
$$x_{4} = 3 + \frac{\sqrt{2} i \sqrt{-1 + \sqrt{41}}}{2}$$
x4 = 3 + sqrt(2)*i*sqrt(-1 + sqrt(41))/2
Sum and product of roots
[src]
sum
____________ ____________ _____________ _____________
___ / ____ ___ / ____ ___ / ____ ___ / ____
\/ 2 *\/ 1 + \/ 41 \/ 2 *\/ 1 + \/ 41 I*\/ 2 *\/ -1 + \/ 41 I*\/ 2 *\/ -1 + \/ 41
3 - --------------------- + 3 + --------------------- + 3 - ------------------------ + 3 + ------------------------
2 2 2 2
$$\left(\left(\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{41}}}{2} + 3\right) + \left(\frac{\sqrt{2} \sqrt{1 + \sqrt{41}}}{2} + 3\right)\right) + \left(3 - \frac{\sqrt{2} i \sqrt{-1 + \sqrt{41}}}{2}\right)\right) + \left(3 + \frac{\sqrt{2} i \sqrt{-1 + \sqrt{41}}}{2}\right)$$
=
12
$$12$$
product
/ ____________\ / ____________\ / _____________\ / _____________\ | ___ / ____ | | ___ / ____ | | ___ / ____ | | ___ / ____ | | \/ 2 *\/ 1 + \/ 41 | | \/ 2 *\/ 1 + \/ 41 | | I*\/ 2 *\/ -1 + \/ 41 | | I*\/ 2 *\/ -1 + \/ 41 | |3 - ---------------------|*|3 + ---------------------|*|3 - ------------------------|*|3 + ------------------------| \ 2 / \ 2 / \ 2 / \ 2 /
$$\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{41}}}{2} + 3\right) \left(\frac{\sqrt{2} \sqrt{1 + \sqrt{41}}}{2} + 3\right) \left(3 - \frac{\sqrt{2} i \sqrt{-1 + \sqrt{41}}}{2}\right) \left(3 + \frac{\sqrt{2} i \sqrt{-1 + \sqrt{41}}}{2}\right)$$
=
62
$$62$$
62
Numerical answer
[src]
x1 = 1.07605558325704
x2 = 3.0 - 1.6436429413703*i
x3 = 3.0 + 1.6436429413703*i
x4 = 4.92394441674296
x4 = 4.92394441674296