Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation -20*(x-13)=-220
-
Equation x-x/12=55/12
-
Equation 3^x=81
-
Equation 12-x^2=11
- Express {x} in terms of y where:
- 18*x+19*y=7
- 16*x-15*y=4
- 1*x+6*y=6
- 20*x+9*y=17
- Canonical form:
- x^2-6x+14
-
Identical expressions
- x^ two -6x+ fourteen = zero
- x squared minus 6x plus 14 equally 0
- x to the power of two minus 6x plus fourteen equally zero
- x2-6x+14=0
- x²-6x+14=0
- x to the power of 2-6x+14=0
- x^2-6x+14=O
-
Similar expressions
- x^2-6x-14=0
- x^2+6x+14=0
x^2-6x+14=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
$$a = 1$$
$$b = -6$$
$$c = 14$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = 3 + \sqrt{5} i$$
$$x_{2} = 3 - \sqrt{5} i$$
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
$$a = 1$$
$$b = -6$$
$$c = 14$$
, then
D = b^2 - 4 * a * c =
(-6)^2 - 4 * (1) * (14) = -20
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
$$x_{1} = 3 + \sqrt{5} i$$
$$x_{2} = 3 - \sqrt{5} i$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -6$$
$$q = \frac{c}{a}$$
$$q = 14$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 6$$
$$x_{1} x_{2} = 14$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -6$$
$$q = \frac{c}{a}$$
$$q = 14$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 6$$
$$x_{1} x_{2} = 14$$
Sum and product of roots
[src]
sum
___ ___ 3 - I*\/ 5 + 3 + I*\/ 5
$$\left(3 - \sqrt{5} i\right) + \left(3 + \sqrt{5} i\right)$$
=
6
$$6$$
product
/ ___\ / ___\ \3 - I*\/ 5 /*\3 + I*\/ 5 /
$$\left(3 - \sqrt{5} i\right) \left(3 + \sqrt{5} i\right)$$
=
14
$$14$$
14
Rapid solution
[src]
___ x1 = 3 - I*\/ 5
$$x_{1} = 3 - \sqrt{5} i$$
___ x2 = 3 + I*\/ 5
$$x_{2} = 3 + \sqrt{5} i$$
x2 = 3 + sqrt(5)*i
Numerical answer
[src]
x1 = 3.0 + 2.23606797749979*i
x2 = 3.0 - 2.23606797749979*i
x2 = 3.0 - 2.23606797749979*i