Mister Exam

Other calculators

x^2-5x+14=0

x^2-5x+14=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  - 5*x + 14 = 0
x2−5x+14=0x^{2} - 5 x + 14 = 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=−5b = -5
c=14c = 14
, then
D = b^2 - 4 * a * c = 

(-5)^2 - 4 * (1) * (14) = -31

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=52+31i2x_{1} = \frac{5}{2} + \frac{\sqrt{31} i}{2}
Simplify
x2=52−31i2x_{2} = \frac{5}{2} - \frac{\sqrt{31} i}{2}
Simplify
Vieta's Theorem
it is reduced quadratic equation
px+q+x2=0p x + q + x^{2} = 0
where
p=bap = \frac{b}{a}
p=−5p = -5
q=caq = \frac{c}{a}
q=14q = 14
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=5x_{1} + x_{2} = 5
x1x2=14x_{1} x_{2} = 14
The graph
0.01.02.03.04.05.06.07.08.0020
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
             ____
     5   I*\/ 31 
x1 = - - --------
     2      2
x1=52−31i2x_{1} = \frac{5}{2} - \frac{\sqrt{31} i}{2}
Simplify 
             ____
     5   I*\/ 31 
x2 = - + --------
     2      2
x2=52+31i2x_{2} = \frac{5}{2} + \frac{\sqrt{31} i}{2}
Simplify
Sum and product of roots [src] 
sum
            ____           ____
    5   I*\/ 31    5   I*\/ 31 
0 + - - -------- + - + --------
    2      2       2      2
(0+(52−31i2))+(52+31i2)\left(0 + \left(\frac{5}{2} - \frac{\sqrt{31} i}{2}\right)\right) + \left(\frac{5}{2} + \frac{\sqrt{31} i}{2}\right) 
=
5
55 
product
  /        ____\ /        ____\
  |5   I*\/ 31 | |5   I*\/ 31 |
1*|- - --------|*|- + --------|
  \2      2    / \2      2    /
1⋅(52−31i2)(52+31i2)1 \cdot \left(\frac{5}{2} - \frac{\sqrt{31} i}{2}\right) \left(\frac{5}{2} + \frac{\sqrt{31} i}{2}\right) 
=
14
1414 
14
Numerical answer [src] 
x1 = 2.5 + 2.78388218141501*i
x2 = 2.5 - 2.78388218141501*i
x2 = 2.5 - 2.78388218141501*i
The graph
x^2-5x+14=0 equation
    © Mister Exam – Calculator