Mister Exam
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How to use it?
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Equation -5x=25
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- Factor polynomial:
- -x^2-x-2
- Graphing y =:
- -x^2-x-2
- Factor squared:
- -x^2-x-2
-
Identical expressions
- -x^ two -x- two
- minus x squared minus x minus 2
- minus x to the power of two minus x minus two
- -x2-x-2
- -x²-x-2
- -x to the power of 2-x-2
-
Similar expressions
- -x^2+x-2
- -x^2-x+2
- x^2-x-2
-x^2-x-2 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
$$a = -1$$
$$b = -1$$
$$c = -2$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \frac{1}{2} - \frac{\sqrt{7} i}{2}$$
$$x_{2} = - \frac{1}{2} + \frac{\sqrt{7} i}{2}$$
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
$$a = -1$$
$$b = -1$$
$$c = -2$$
, 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.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{1}{2} - \frac{\sqrt{7} i}{2}$$
$$x_{2} = - \frac{1}{2} + \frac{\sqrt{7} i}{2}$$
Vieta's Theorem
rewrite the equation
$$\left(- x^{2} - x\right) - 2 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + x + 2 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = 2$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -1$$
$$x_{1} x_{2} = 2$$
$$\left(- x^{2} - x\right) - 2 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + x + 2 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = 2$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -1$$
$$x_{1} x_{2} = 2$$
Rapid solution
[src]
___
1 I*\/ 7
x1 = - - - -------
2 2
$$x_{1} = - \frac{1}{2} - \frac{\sqrt{7} i}{2}$$
___
1 I*\/ 7
x2 = - - + -------
2 2
$$x_{2} = - \frac{1}{2} + \frac{\sqrt{7} i}{2}$$
x2 = -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]
x1 = -0.5 + 1.3228756555323*i
x2 = -0.5 - 1.3228756555323*i
x2 = -0.5 - 1.3228756555323*i