Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation 4x+2(1-3x)=13
-
Equation x^2-10x+25=0
- Equation x-y=0
-
Equation 3-x=1+x
- Express {x} in terms of y where:
- -2*x-4*y=11
- 14*x-3*y=-9
- 6*x-19*y=-18
- 10*x-5*y=3
- Derivative of:
-
x^2+3*x+5
- Factor squared:
- x^2+3*x+5
-
Identical expressions
- x^ two + three *x+ five = zero
- x squared plus 3 multiply by x plus 5 equally 0
- x to the power of two plus three multiply by x plus five equally zero
- x2+3*x+5=0
- x²+3*x+5=0
- x to the power of 2+3*x+5=0
- x^2+3x+5=0
- x2+3x+5=0
- x^2+3*x+5=O
-
Similar expressions
- x^2-3*x+5=0
- x^2+3*x-5=0
x^2+3*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
$$a = 1$$
$$b = 3$$
$$c = 5$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \frac{3}{2} + \frac{\sqrt{11} i}{2}$$
$$x_{2} = - \frac{3}{2} - \frac{\sqrt{11} i}{2}$$
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
$$a = 1$$
$$b = 3$$
$$c = 5$$
, then
D = b^2 - 4 * a * c =
(3)^2 - 4 * (1) * (5) = -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
$$x_{1} = - \frac{3}{2} + \frac{\sqrt{11} i}{2}$$
$$x_{2} = - \frac{3}{2} - \frac{\sqrt{11} i}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 3$$
$$q = \frac{c}{a}$$
$$q = 5$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -3$$
$$x_{1} x_{2} = 5$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 3$$
$$q = \frac{c}{a}$$
$$q = 5$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -3$$
$$x_{1} x_{2} = 5$$
Sum and product of roots
[src]
sum
____ ____ 3 I*\/ 11 3 I*\/ 11 - - - -------- + - - + -------- 2 2 2 2
$$\left(- \frac{3}{2} - \frac{\sqrt{11} i}{2}\right) + \left(- \frac{3}{2} + \frac{\sqrt{11} i}{2}\right)$$
=
-3
$$-3$$
product
/ ____\ / ____\ | 3 I*\/ 11 | | 3 I*\/ 11 | |- - - --------|*|- - + --------| \ 2 2 / \ 2 2 /
$$\left(- \frac{3}{2} - \frac{\sqrt{11} i}{2}\right) \left(- \frac{3}{2} + \frac{\sqrt{11} i}{2}\right)$$
=
5
$$5$$
5
Rapid solution
[src]
____
3 I*\/ 11
x1 = - - - --------
2 2
$$x_{1} = - \frac{3}{2} - \frac{\sqrt{11} i}{2}$$
____
3 I*\/ 11
x2 = - - + --------
2 2
$$x_{2} = - \frac{3}{2} + \frac{\sqrt{11} i}{2}$$
x2 = -3/2 + sqrt(11)*i/2
Numerical answer
[src]
x1 = -1.5 + 1.6583123951777*i
x2 = -1.5 - 1.6583123951777*i
x2 = -1.5 - 1.6583123951777*i
The graph