Mister Exam
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How to use it?
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Equation laplace(x)=(39/40)
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Identical expressions
- t^ two -t+ two = zero
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- t to the power of two minus t plus two equally zero
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- t^2-t+2=O
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Similar expressions
- t^2-t-2=0
- t^2+t+2=0
t^2-t+2=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:
$$t_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$t_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -1$$
$$c = 2$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$t_{1} = \frac{1}{2} + \frac{\sqrt{7} i}{2}$$
$$t_{2} = \frac{1}{2} - \frac{\sqrt{7} i}{2}$$
a*t^2 + b*t + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$t_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$t_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -1$$
$$c = 2$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1) * (2) = -7
Because D<0, then the equation
has no real roots,
but complex roots is exists.
t1 = (-b + sqrt(D)) / (2*a)
t2 = (-b - sqrt(D)) / (2*a)
or
$$t_{1} = \frac{1}{2} + \frac{\sqrt{7} i}{2}$$
$$t_{2} = \frac{1}{2} - \frac{\sqrt{7} i}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p t + q + t^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 2$$
Vieta Formulas
$$t_{1} + t_{2} = - p$$
$$t_{1} t_{2} = q$$
$$t_{1} + t_{2} = 1$$
$$t_{1} t_{2} = 2$$
$$p t + q + t^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 2$$
Vieta Formulas
$$t_{1} + t_{2} = - p$$
$$t_{1} t_{2} = q$$
$$t_{1} + t_{2} = 1$$
$$t_{1} t_{2} = 2$$
Rapid solution
[src]
___
1 I*\/ 7
t1 = - - -------
2 2
$$t_{1} = \frac{1}{2} - \frac{\sqrt{7} i}{2}$$
___
1 I*\/ 7
t2 = - + -------
2 2
$$t_{2} = \frac{1}{2} + \frac{\sqrt{7} i}{2}$$
t2 = 1/2 + sqrt(7)*i/2
Sum and product of roots
[src]
sum
___ ___ 1 I*\/ 7 1 I*\/ 7 - - ------- + - + ------- 2 2 2 2
$$\left(\frac{1}{2} - \frac{\sqrt{7} i}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{7} i}{2}\right)$$
=
1
$$1$$
product
/ ___\ / ___\ |1 I*\/ 7 | |1 I*\/ 7 | |- - -------|*|- + -------| \2 2 / \2 2 /
$$\left(\frac{1}{2} - \frac{\sqrt{7} i}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{7} i}{2}\right)$$
=
2
$$2$$
2
Numerical answer
[src]
t1 = 0.5 - 1.3228756555323*i
t2 = 0.5 + 1.3228756555323*i
t2 = 0.5 + 1.3228756555323*i
The graph