Mister Exam
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How to use it?
- Equation:
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Equation 3*x^2+x-10=0
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Equation 2x=5
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Equation 5^(9-x)=64
- Express {x} in terms of y where:
- -8*x+14*y=5
- 17*x+7*y=-6
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- Factor squared:
- x^2+x+5
-
Identical expressions
- x^ two +x+ five = zero
- x squared plus x plus 5 equally 0
- x to the power of two plus x plus five equally zero
- x2+x+5=0
- x²+x+5=0
- x to the power of 2+x+5=0
- x^2+x+5=O
-
Similar expressions
- x^2-x+5=0
- x^2+x-5=0
x^2+x+5=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
$$a = 1$$
$$b = 1$$
$$c = 5$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \frac{1}{2} + \frac{\sqrt{19} i}{2}$$
$$x_{2} = - \frac{1}{2} - \frac{\sqrt{19} i}{2}$$
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
$$a = 1$$
$$b = 1$$
$$c = 5$$
, then
D = b^2 - 4 * a * c =
(1)^2 - 4 * (1) * (5) = -19
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
$$x_{1} = - \frac{1}{2} + \frac{\sqrt{19} i}{2}$$
$$x_{2} = - \frac{1}{2} - \frac{\sqrt{19} i}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = 5$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -1$$
$$x_{1} x_{2} = 5$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = 5$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -1$$
$$x_{1} x_{2} = 5$$
Sum and product of roots
[src]
sum
____ ____ 1 I*\/ 19 1 I*\/ 19 - - - -------- + - - + -------- 2 2 2 2
$$\left(- \frac{1}{2} - \frac{\sqrt{19} i}{2}\right) + \left(- \frac{1}{2} + \frac{\sqrt{19} i}{2}\right)$$
=
-1
$$-1$$
product
/ ____\ / ____\ | 1 I*\/ 19 | | 1 I*\/ 19 | |- - - --------|*|- - + --------| \ 2 2 / \ 2 2 /
$$\left(- \frac{1}{2} - \frac{\sqrt{19} i}{2}\right) \left(- \frac{1}{2} + \frac{\sqrt{19} i}{2}\right)$$
=
5
$$5$$
5
Rapid solution
[src]
____
1 I*\/ 19
x1 = - - - --------
2 2
$$x_{1} = - \frac{1}{2} - \frac{\sqrt{19} i}{2}$$
____
1 I*\/ 19
x2 = - - + --------
2 2
$$x_{2} = - \frac{1}{2} + \frac{\sqrt{19} i}{2}$$
x2 = -1/2 + sqrt(19)*i/2
Numerical answer
[src]
x1 = -0.5 + 2.17944947177034*i
x2 = -0.5 - 2.17944947177034*i
x2 = -0.5 - 2.17944947177034*i
The graph