Mister Exam

Other calculators

3x^2-4x+5 equation

The teacher will be very surprised to see your correct solution 😉

v

Numerical solution:

Do search numerical solution at [, ]

The solution

You have entered [src]
   2              
3*x  - 4*x + 5 = 0
$$\left(3 x^{2} - 4 x\right) + 5 = 0$$
Detail solution
This equation is of the form
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
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{4}{3}$$
$$x_{1} x_{2} = \frac{5}{3}$$
The graph
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