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Identical expressions
- two *x^ two / ten ^ four + nine *x+ thirty thousand = zero
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Similar expressions
- 2*x^2/10^4+9*x-30000=0
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2*x^2/10^4+9*x+30000=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
2 2*x ----- + 9*x + 30000 = 0 10000
$$\left(9 x + \frac{2 x^{2}}{10000}\right) + 30000 = 0$$
Detail solution
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
$$a = \frac{1}{5000}$$
$$b = 9$$
$$c = 30000$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = -22500 + 2500 \sqrt{57}$$
$$x_{2} = -22500 - 2500 \sqrt{57}$$
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
$$a = \frac{1}{5000}$$
$$b = 9$$
$$c = 30000$$
, then
D = b^2 - 4 * a * c =
(9)^2 - 4 * (1/5000) * (30000) = 57
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = -22500 + 2500 \sqrt{57}$$
$$x_{2} = -22500 - 2500 \sqrt{57}$$
Vieta's Theorem
rewrite the equation
$$\left(9 x + \frac{2 x^{2}}{10000}\right) + 30000 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + 45000 x + 150000000 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 45000$$
$$q = \frac{c}{a}$$
$$q = 150000000$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -45000$$
$$x_{1} x_{2} = 150000000$$
$$\left(9 x + \frac{2 x^{2}}{10000}\right) + 30000 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + 45000 x + 150000000 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 45000$$
$$q = \frac{c}{a}$$
$$q = 150000000$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -45000$$
$$x_{1} x_{2} = 150000000$$
Rapid solution
[src]
____ x1 = -22500 - 2500*\/ 57
$$x_{1} = -22500 - 2500 \sqrt{57}$$
____ x2 = -22500 + 2500*\/ 57
$$x_{2} = -22500 + 2500 \sqrt{57}$$
x2 = -22500 + 2500*sqrt(57)
Sum and product of roots
[src]
sum
____ ____ -22500 - 2500*\/ 57 + -22500 + 2500*\/ 57
$$\left(-22500 - 2500 \sqrt{57}\right) + \left(-22500 + 2500 \sqrt{57}\right)$$
=
-45000
$$-45000$$
product
/ ____\ / ____\ \-22500 - 2500*\/ 57 /*\-22500 + 2500*\/ 57 /
$$\left(-22500 - 2500 \sqrt{57}\right) \left(-22500 + 2500 \sqrt{57}\right)$$
=
150000000
$$150000000$$
150000000