Mister Exam
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How to use it?
- Equation:
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Equation 5*x^2+25*x=0
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Equation -2x+16=5x+30
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- Factor squared:
- x^2+3*x+3
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Identical expressions
- x^ two + three *x+ three = zero
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Similar expressions
- x^2-3*x+3=0
- x^2+3*x-3=0
x^2+3*x+3=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
$$a = 1$$
$$b = 3$$
$$c = 3$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \frac{3}{2} + \frac{\sqrt{3} i}{2}$$
$$x_{2} = - \frac{3}{2} - \frac{\sqrt{3} i}{2}$$
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
$$a = 1$$
$$b = 3$$
$$c = 3$$
, then
D = b^2 - 4 * a * c =
(3)^2 - 4 * (1) * (3) = -3
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
$$x_{1} = - \frac{3}{2} + \frac{\sqrt{3} i}{2}$$
$$x_{2} = - \frac{3}{2} - \frac{\sqrt{3} i}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 3$$
$$q = \frac{c}{a}$$
$$q = 3$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -3$$
$$x_{1} x_{2} = 3$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 3$$
$$q = \frac{c}{a}$$
$$q = 3$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -3$$
$$x_{1} x_{2} = 3$$
Rapid solution
[src]
___
3 I*\/ 3
x1 = - - - -------
2 2
$$x_{1} = - \frac{3}{2} - \frac{\sqrt{3} i}{2}$$
___
3 I*\/ 3
x2 = - - + -------
2 2
$$x_{2} = - \frac{3}{2} + \frac{\sqrt{3} i}{2}$$
x2 = -3/2 + sqrt(3)*i/2
Sum and product of roots
[src]
sum
___ ___ 3 I*\/ 3 3 I*\/ 3 - - - ------- + - - + ------- 2 2 2 2
$$\left(- \frac{3}{2} - \frac{\sqrt{3} i}{2}\right) + \left(- \frac{3}{2} + \frac{\sqrt{3} i}{2}\right)$$
=
-3
$$-3$$
product
/ ___\ / ___\ | 3 I*\/ 3 | | 3 I*\/ 3 | |- - - -------|*|- - + -------| \ 2 2 / \ 2 2 /
$$\left(- \frac{3}{2} - \frac{\sqrt{3} i}{2}\right) \left(- \frac{3}{2} + \frac{\sqrt{3} i}{2}\right)$$
=
3
$$3$$
3
Numerical answer
[src]
x1 = -1.5 - 0.866025403784439*i
x2 = -1.5 + 0.866025403784439*i
x2 = -1.5 + 0.866025403784439*i
The graph