Mister Exam
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How to use it?
- Equation:
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Equation x^2-6x=0
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Equation x*x-x+7=0
- Express {x} in terms of y where:
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Identical expressions
- x*x-x+ seven = zero
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Similar expressions
- x*x-x-7=0
- x*x+x+7=0
x*x-x+7=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
$$a = 1$$
$$b = -1$$
$$c = 7$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{1}{2} + \frac{3 \sqrt{3} i}{2}$$
$$x_{2} = \frac{1}{2} - \frac{3 \sqrt{3} i}{2}$$
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
$$a = 1$$
$$b = -1$$
$$c = 7$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1) * (7) = -27
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
$$x_{1} = \frac{1}{2} + \frac{3 \sqrt{3} i}{2}$$
$$x_{2} = \frac{1}{2} - \frac{3 \sqrt{3} i}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 7$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = 7$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 7$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = 7$$
Rapid solution
[src]
___
1 3*I*\/ 3
x1 = - - ---------
2 2
$$x_{1} = \frac{1}{2} - \frac{3 \sqrt{3} i}{2}$$
___
1 3*I*\/ 3
x2 = - + ---------
2 2
$$x_{2} = \frac{1}{2} + \frac{3 \sqrt{3} i}{2}$$
x2 = 1/2 + 3*sqrt(3)*i/2
Sum and product of roots
[src]
sum
___ ___ 1 3*I*\/ 3 1 3*I*\/ 3 - - --------- + - + --------- 2 2 2 2
$$\left(\frac{1}{2} - \frac{3 \sqrt{3} i}{2}\right) + \left(\frac{1}{2} + \frac{3 \sqrt{3} i}{2}\right)$$
=
1
$$1$$
product
/ ___\ / ___\ |1 3*I*\/ 3 | |1 3*I*\/ 3 | |- - ---------|*|- + ---------| \2 2 / \2 2 /
$$\left(\frac{1}{2} - \frac{3 \sqrt{3} i}{2}\right) \left(\frac{1}{2} + \frac{3 \sqrt{3} i}{2}\right)$$
=
7
$$7$$
7
Numerical answer
[src]
x1 = 0.5 + 2.59807621135332*i
x2 = 0.5 - 2.59807621135332*i
x2 = 0.5 - 2.59807621135332*i
The graph