Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 1/(x-3)^2-3/(x-3)-4=0
-
Equation x^4+2*x^2-8=0
-
Equation x^2-2x+1=0
-
Equation x^4+4=0
- Express {x} in terms of y where:
- 17*x-20*y=6
- 5*x-13*y=17
- 6*x+17*y=3
- 1*x+4*y=20
- Graphing y =:
-
3x^2-4x+5
- Integral of d{x}:
- 3x^2-4x+5
- Derivative of:
-
3x^2-4x+5
-
Identical expressions
- 3x^ two -4x+ five
- 3x squared minus 4x plus 5
- 3x to the power of two minus 4x plus five
- 3x2-4x+5
- 3x²-4x+5
- 3x to the power of 2-4x+5
-
Similar expressions
- 3x^2-4x-5
- 3x^2+4x+5
3x^2-4x+5 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
$$a = 3$$
$$b = -4$$
$$c = 5$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{2}{3} + \frac{\sqrt{11} i}{3}$$
$$x_{2} = \frac{2}{3} - \frac{\sqrt{11} i}{3}$$
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
$$a = 3$$
$$b = -4$$
$$c = 5$$
, then
D = b^2 - 4 * a * c =
(-4)^2 - 4 * (3) * (5) = -44
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
$$x_{1} = \frac{2}{3} + \frac{\sqrt{11} i}{3}$$
$$x_{2} = \frac{2}{3} - \frac{\sqrt{11} i}{3}$$
Vieta's Theorem
rewrite the equation
$$\left(3 x^{2} - 4 x\right) + 5 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{4 x}{3} + \frac{5}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{4}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{5}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{4}{3}$$
$$x_{1} x_{2} = \frac{5}{3}$$
$$\left(3 x^{2} - 4 x\right) + 5 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{4 x}{3} + \frac{5}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{4}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{5}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{4}{3}$$
$$x_{1} x_{2} = \frac{5}{3}$$
Rapid solution
[src]
____
2 I*\/ 11
x1 = - - --------
3 3
$$x_{1} = \frac{2}{3} - \frac{\sqrt{11} i}{3}$$
____
2 I*\/ 11
x2 = - + --------
3 3
$$x_{2} = \frac{2}{3} + \frac{\sqrt{11} i}{3}$$
x2 = 2/3 + sqrt(11)*i/3
Sum and product of roots
[src]
sum
____ ____ 2 I*\/ 11 2 I*\/ 11 - - -------- + - + -------- 3 3 3 3
$$\left(\frac{2}{3} - \frac{\sqrt{11} i}{3}\right) + \left(\frac{2}{3} + \frac{\sqrt{11} i}{3}\right)$$
=
4/3
$$\frac{4}{3}$$
product
/ ____\ / ____\ |2 I*\/ 11 | |2 I*\/ 11 | |- - --------|*|- + --------| \3 3 / \3 3 /
$$\left(\frac{2}{3} - \frac{\sqrt{11} i}{3}\right) \left(\frac{2}{3} + \frac{\sqrt{11} i}{3}\right)$$
=
5/3
$$\frac{5}{3}$$
5/3
Numerical answer
[src]
x1 = 0.666666666666667 + 1.10554159678513*i
x2 = 0.666666666666667 - 1.10554159678513*i
x2 = 0.666666666666667 - 1.10554159678513*i