Mister Exam

Other calculators

x^2-3x+5=0

x^2-3x+5=0 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              
x  - 3*x + 5 = 0
(x2−3x)+5=0\left(x^{2} - 3 x\right) + 5 = 0
Simplify
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:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
where D = b^2 - 4*a*c - it is the discriminant.
Because
a=1a = 1
b=−3b = -3
c=5c = 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
x1=32+11i2x_{1} = \frac{3}{2} + \frac{\sqrt{11} i}{2}
x2=32−11i2x_{2} = \frac{3}{2} - \frac{\sqrt{11} i}{2}
Vieta's Theorem
it is reduced quadratic equation
px+q+x2=0p x + q + x^{2} = 0
where
p=bap = \frac{b}{a}
p=−3p = -3
q=caq = \frac{c}{a}
q=5q = 5
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=3x_{1} + x_{2} = 3
x1x2=5x_{1} x_{2} = 5
The graph
012345678-3-2-1020
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
             ____
     3   I*\/ 11 
x1 = - - --------
     2      2
x1=32−11i2x_{1} = \frac{3}{2} - \frac{\sqrt{11} i}{2} 
             ____
     3   I*\/ 11 
x2 = - + --------
     2      2
x2=32+11i2x_{2} = \frac{3}{2} + \frac{\sqrt{11} i}{2} 
x2 = 3/2 + sqrt(11)*i/2
Sum and product of roots [src] 
sum
        ____           ____
3   I*\/ 11    3   I*\/ 11 
- - -------- + - + --------
2      2       2      2
(32−11i2)+(32+11i2)\left(\frac{3}{2} - \frac{\sqrt{11} i}{2}\right) + \left(\frac{3}{2} + \frac{\sqrt{11} i}{2}\right) 
=
3
33 
product
/        ____\ /        ____\
|3   I*\/ 11 | |3   I*\/ 11 |
|- - --------|*|- + --------|
\2      2    / \2      2    /
(32−11i2)(32+11i2)\left(\frac{3}{2} - \frac{\sqrt{11} i}{2}\right) \left(\frac{3}{2} + \frac{\sqrt{11} i}{2}\right) 
=
5
55 
5
Numerical answer [src] 
x1 = 1.5 - 1.6583123951777*i
x2 = 1.5 + 1.6583123951777*i
x2 = 1.5 + 1.6583123951777*i
The graph
x^2-3x+5=0 equation
    © Mister Exam – Calculator