Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation (x+2)^4-4*(x+2)^2-5=0
-
Equation 12/(x+3)=-6/7
-
Equation x-6/2-x/3=3
-
Equation x^3+2x^2-x-2=0
- Express {x} in terms of y where:
- 4*x+8*y=7
- -14*x+5*y=3
- 13*x-18*y=-17
- 9*x-1*y=17
- Derivative of:
-
x^2-x+3
- Factor squared:
- x^2-x+3
- Graphing y =:
- x^2-x+3
-
Identical expressions
- x^ two -x+ three = zero
- x squared minus x plus 3 equally 0
- x to the power of two minus x plus three equally zero
- x2-x+3=0
- x²-x+3=0
- x to the power of 2-x+3=0
- x^2-x+3=O
-
Similar expressions
- x^2+x+3=0
- x^2-x-3=0
x^2-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
$$a = 1$$
$$b = -1$$
$$c = 3$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{1}{2} + \frac{\sqrt{11} i}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{11} i}{2}$$
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
$$a = 1$$
$$b = -1$$
$$c = 3$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1) * (3) = -11
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
$$x_{1} = \frac{1}{2} + \frac{\sqrt{11} i}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{11} i}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 3$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = 3$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 3$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = 3$$
Sum and product of roots
[src]
sum
____ ____ 1 I*\/ 11 1 I*\/ 11 - - -------- + - + -------- 2 2 2 2
$$\left(\frac{1}{2} - \frac{\sqrt{11} i}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{11} i}{2}\right)$$
=
1
$$1$$
product
/ ____\ / ____\ |1 I*\/ 11 | |1 I*\/ 11 | |- - --------|*|- + --------| \2 2 / \2 2 /
$$\left(\frac{1}{2} - \frac{\sqrt{11} i}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{11} i}{2}\right)$$
=
3
$$3$$
3
Rapid solution
[src]
____
1 I*\/ 11
x1 = - - --------
2 2
$$x_{1} = \frac{1}{2} - \frac{\sqrt{11} i}{2}$$
____
1 I*\/ 11
x2 = - + --------
2 2
$$x_{2} = \frac{1}{2} + \frac{\sqrt{11} i}{2}$$
x2 = 1/2 + sqrt(11)*i/2
Numerical answer
[src]
x1 = 0.5 + 1.6583123951777*i
x2 = 0.5 - 1.6583123951777*i
x2 = 0.5 - 1.6583123951777*i
The graph