Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation sin(x)=4
-
Equation 5^x=125
-
Equation (|x+4|)=5
-
Equation 2*sin(x)=1
- Express {x} in terms of y where:
- 16*x-17*y=-15
- -4*x-4*y=19
- 15*x+1*y=19
- -5*x-2*y=9
-
Identical expressions
- x^ two + six *x+ nineteen = zero
- x squared plus 6 multiply by x plus 19 equally 0
- x to the power of two plus six multiply by x plus nineteen equally zero
- x2+6*x+19=0
- x²+6*x+19=0
- x to the power of 2+6*x+19=0
- x^2+6x+19=0
- x2+6x+19=0
- x^2+6*x+19=O
-
Similar expressions
- x^2-6*x+19=0
- x^2+6*x-19=0
x^2+6*x+19=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
$$a = 1$$
$$b = 6$$
$$c = 19$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = -3 + \sqrt{10} i$$
$$x_{2} = -3 - \sqrt{10} i$$
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
$$a = 1$$
$$b = 6$$
$$c = 19$$
, then
D = b^2 - 4 * a * c =
(6)^2 - 4 * (1) * (19) = -40
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
$$x_{1} = -3 + \sqrt{10} i$$
$$x_{2} = -3 - \sqrt{10} i$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 6$$
$$q = \frac{c}{a}$$
$$q = 19$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -6$$
$$x_{1} x_{2} = 19$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 6$$
$$q = \frac{c}{a}$$
$$q = 19$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -6$$
$$x_{1} x_{2} = 19$$
Sum and product of roots
[src]
sum
____ ____ -3 - I*\/ 10 + -3 + I*\/ 10
$$\left(-3 - \sqrt{10} i\right) + \left(-3 + \sqrt{10} i\right)$$
=
-6
$$-6$$
product
/ ____\ / ____\ \-3 - I*\/ 10 /*\-3 + I*\/ 10 /
$$\left(-3 - \sqrt{10} i\right) \left(-3 + \sqrt{10} i\right)$$
=
19
$$19$$
19
Rapid solution
[src]
____ x1 = -3 - I*\/ 10
$$x_{1} = -3 - \sqrt{10} i$$
____ x2 = -3 + I*\/ 10
$$x_{2} = -3 + \sqrt{10} i$$
x2 = -3 + sqrt(10)*i
Numerical answer
[src]
x1 = -3.0 + 3.16227766016838*i
x2 = -3.0 - 3.16227766016838*i
x2 = -3.0 - 3.16227766016838*i