Mister Exam
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x^2-20*x+102=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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
$$a = 1$$
$$b = -20$$
$$c = 102$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = 10 + \sqrt{2} i$$
$$x_{2} = 10 - \sqrt{2} i$$
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
$$a = 1$$
$$b = -20$$
$$c = 102$$
, then
D = b^2 - 4 * a * c =
(-20)^2 - 4 * (1) * (102) = -8
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
$$x_{1} = 10 + \sqrt{2} i$$
$$x_{2} = 10 - \sqrt{2} i$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -20$$
$$q = \frac{c}{a}$$
$$q = 102$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 20$$
$$x_{1} x_{2} = 102$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -20$$
$$q = \frac{c}{a}$$
$$q = 102$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 20$$
$$x_{1} x_{2} = 102$$
Rapid solution
[src]
___ x1 = 10 - I*\/ 2
$$x_{1} = 10 - \sqrt{2} i$$
___ x2 = 10 + I*\/ 2
$$x_{2} = 10 + \sqrt{2} i$$
x2 = 10 + sqrt(2)*i
Sum and product of roots
[src]
sum
___ ___ 10 - I*\/ 2 + 10 + I*\/ 2
$$\left(10 - \sqrt{2} i\right) + \left(10 + \sqrt{2} i\right)$$
=
20
$$20$$
product
/ ___\ / ___\ \10 - I*\/ 2 /*\10 + I*\/ 2 /
$$\left(10 - \sqrt{2} i\right) \left(10 + \sqrt{2} i\right)$$
=
102
$$102$$
102
Numerical answer
[src]
x1 = 10.0 - 1.4142135623731*i
x2 = 10.0 + 1.4142135623731*i
x2 = 10.0 + 1.4142135623731*i