Mister Exam
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- three *x^ four + nine *x^ two - thirty = zero
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Similar expressions
- 3*x^4+9*x^2+30=0
- 3*x^4-9*x^2-30=0
3*x^4+9*x^2-30=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$\left(3 x^{4} + 9 x^{2}\right) - 30 = 0$$
Do replacement
$$v = x^{2}$$
then the equation will be the:
$$3 v^{2} + 9 v - 30 = 0$$
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
$$a = 3$$
$$b = 9$$
$$c = -30$$
, then
Because D > 0, then the equation has two roots.
or
$$v_{1} = 2$$
$$v_{2} = -5$$
The final answer:
Because
$$v = x^{2}$$
then
$$x_{1} = \sqrt{v_{1}}$$
$$x_{2} = - \sqrt{v_{1}}$$
$$x_{3} = \sqrt{v_{2}}$$
$$x_{4} = - \sqrt{v_{2}}$$
then:
$$x_{1} = $$
$$\frac{0}{1} + \frac{2^{\frac{1}{2}}}{1} = \sqrt{2}$$
$$x_{2} = $$
$$\frac{\left(-1\right) 2^{\frac{1}{2}}}{1} + \frac{0}{1} = - \sqrt{2}$$
$$x_{3} = $$
$$\frac{0}{1} + \frac{\left(-5\right)^{\frac{1}{2}}}{1} = \sqrt{5} i$$
$$x_{4} = $$
$$\frac{0}{1} + \frac{\left(-1\right) \left(-5\right)^{\frac{1}{2}}}{1} = - \sqrt{5} i$$
$$\left(3 x^{4} + 9 x^{2}\right) - 30 = 0$$
Do replacement
$$v = x^{2}$$
then the equation will be the:
$$3 v^{2} + 9 v - 30 = 0$$
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
$$a = 3$$
$$b = 9$$
$$c = -30$$
, then
D = b^2 - 4 * a * c =
(9)^2 - 4 * (3) * (-30) = 441
Because D > 0, then the equation has two roots.
v1 = (-b + sqrt(D)) / (2*a)
v2 = (-b - sqrt(D)) / (2*a)
or
$$v_{1} = 2$$
$$v_{2} = -5$$
The final answer:
Because
$$v = x^{2}$$
then
$$x_{1} = \sqrt{v_{1}}$$
$$x_{2} = - \sqrt{v_{1}}$$
$$x_{3} = \sqrt{v_{2}}$$
$$x_{4} = - \sqrt{v_{2}}$$
then:
$$x_{1} = $$
$$\frac{0}{1} + \frac{2^{\frac{1}{2}}}{1} = \sqrt{2}$$
$$x_{2} = $$
$$\frac{\left(-1\right) 2^{\frac{1}{2}}}{1} + \frac{0}{1} = - \sqrt{2}$$
$$x_{3} = $$
$$\frac{0}{1} + \frac{\left(-5\right)^{\frac{1}{2}}}{1} = \sqrt{5} i$$
$$x_{4} = $$
$$\frac{0}{1} + \frac{\left(-1\right) \left(-5\right)^{\frac{1}{2}}}{1} = - \sqrt{5} i$$
Rapid solution
[src]
___ x1 = -\/ 2
$$x_{1} = - \sqrt{2}$$
___ x2 = \/ 2
$$x_{2} = \sqrt{2}$$
___ x3 = -I*\/ 5
$$x_{3} = - \sqrt{5} i$$
___ x4 = I*\/ 5
$$x_{4} = \sqrt{5} i$$
x4 = sqrt(5)*i
Sum and product of roots
[src]
sum
___ ___ ___ ___ - \/ 2 + \/ 2 - I*\/ 5 + I*\/ 5
$$\left(\left(- \sqrt{2} + \sqrt{2}\right) - \sqrt{5} i\right) + \sqrt{5} i$$
=
0
$$0$$
product
___ ___ / ___\ ___ -\/ 2 *\/ 2 *\-I*\/ 5 /*I*\/ 5
$$\sqrt{5} i - \sqrt{2} \sqrt{2} \left(- \sqrt{5} i\right)$$
=
-10
$$-10$$
-10
Numerical answer
[src]
x1 = 1.4142135623731
x2 = -1.4142135623731
x3 = -2.23606797749979*i
x4 = 2.23606797749979*i
x4 = 2.23606797749979*i