Mister Exam

Other calculators

7x^2-5x+c=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              
7*x  - 5*x + c = 0
c+(7x2−5x)=0c + \left(7 x^{2} - 5 x\right) = 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=7a = 7
b=−5b = -5
True

, then
D = b^2 - 4 * a * c = 

(-5)^2 - 4 * (7) * (c) = 25 - 28*c

The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
x1=25−28c14+514x_{1} = \frac{\sqrt{25 - 28 c}}{14} + \frac{5}{14}
x2=514−25−28c14x_{2} = \frac{5}{14} - \frac{\sqrt{25 - 28 c}}{14}
Vieta's Theorem
rewrite the equation
c+(7x2−5x)=0c + \left(7 x^{2} - 5 x\right) = 0
of
ax2+bx+c=0a x^{2} + b x + c = 0
as reduced quadratic equation
x2+bxa+ca=0x^{2} + \frac{b x}{a} + \frac{c}{a} = 0
c7+x2−5x7=0\frac{c}{7} + x^{2} - \frac{5 x}{7} = 0
px+q+x2=0p x + q + x^{2} = 0
where
p=bap = \frac{b}{a}
p=−57p = - \frac{5}{7}
q=caq = \frac{c}{a}
q=c7q = \frac{c}{7}
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=57x_{1} + x_{2} = \frac{5}{7}
x1x2=c7x_{1} x_{2} = \frac{c}{7}
The graph
Rapid solution [src] 
             _______________________________                                             _______________________________                                     
          4 /                2         2        /atan2(-28*im(c), 25 - 28*re(c))\     4 /                2         2        /atan2(-28*im(c), 25 - 28*re(c))\
          \/  (25 - 28*re(c))  + 784*im (c) *cos|-------------------------------|   I*\/  (25 - 28*re(c))  + 784*im (c) *sin|-------------------------------|
     5                                          \               2               /                                           \               2               /
x1 = -- - ----------------------------------------------------------------------- - -------------------------------------------------------------------------
     14                                      14                                                                         14
x1=−i(25−28re⁡(c))2+784(im⁡(c))24sin⁡(atan2⁡(−28im⁡(c),25−28re⁡(c))2)14−(25−28re⁡(c))2+784(im⁡(c))24cos⁡(atan2⁡(−28im⁡(c),25−28re⁡(c))2)14+514x_{1} = - \frac{i \sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{5}{14} 
             _______________________________                                             _______________________________                                     
          4 /                2         2        /atan2(-28*im(c), 25 - 28*re(c))\     4 /                2         2        /atan2(-28*im(c), 25 - 28*re(c))\
          \/  (25 - 28*re(c))  + 784*im (c) *cos|-------------------------------|   I*\/  (25 - 28*re(c))  + 784*im (c) *sin|-------------------------------|
     5                                          \               2               /                                           \               2               /
x2 = -- + ----------------------------------------------------------------------- + -------------------------------------------------------------------------
     14                                      14                                                                         14
x2=i(25−28re⁡(c))2+784(im⁡(c))24sin⁡(atan2⁡(−28im⁡(c),25−28re⁡(c))2)14+(25−28re⁡(c))2+784(im⁡(c))24cos⁡(atan2⁡(−28im⁡(c),25−28re⁡(c))2)14+514x_{2} = \frac{i \sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{5}{14} 
x2 = i*((25 - 28*re(c))^2 + 784*im(c)^2)^(1/4)*sin(atan2(-28*im(c, 25 - 28*re(c))/2)/14 + ((25 - 28*re(c))^2 + 784*im(c)^2)^(1/4)*cos(atan2(-28*im(c), 25 - 28*re(c))/2)/14 + 5/14)
Sum and product of roots [src] 
sum
        _______________________________                                             _______________________________                                                _______________________________                                             _______________________________                                     
     4 /                2         2        /atan2(-28*im(c), 25 - 28*re(c))\     4 /                2         2        /atan2(-28*im(c), 25 - 28*re(c))\        4 /                2         2        /atan2(-28*im(c), 25 - 28*re(c))\     4 /                2         2        /atan2(-28*im(c), 25 - 28*re(c))\
     \/  (25 - 28*re(c))  + 784*im (c) *cos|-------------------------------|   I*\/  (25 - 28*re(c))  + 784*im (c) *sin|-------------------------------|        \/  (25 - 28*re(c))  + 784*im (c) *cos|-------------------------------|   I*\/  (25 - 28*re(c))  + 784*im (c) *sin|-------------------------------|
5                                          \               2               /                                           \               2               /   5                                          \               2               /                                           \               2               /
-- - ----------------------------------------------------------------------- - ------------------------------------------------------------------------- + -- + ----------------------------------------------------------------------- + -------------------------------------------------------------------------
14                                      14                                                                         14                                      14                                      14                                                                         14
(−i(25−28re⁡(c))2+784(im⁡(c))24sin⁡(atan2⁡(−28im⁡(c),25−28re⁡(c))2)14−(25−28re⁡(c))2+784(im⁡(c))24cos⁡(atan2⁡(−28im⁡(c),25−28re⁡(c))2)14+514)+(i(25−28re⁡(c))2+784(im⁡(c))24sin⁡(atan2⁡(−28im⁡(c),25−28re⁡(c))2)14+(25−28re⁡(c))2+784(im⁡(c))24cos⁡(atan2⁡(−28im⁡(c),25−28re⁡(c))2)14+514)\left(- \frac{i \sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{5}{14}\right) + \left(\frac{i \sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{5}{14}\right) 
=
5/7
57\frac{5}{7} 
product
/        _______________________________                                             _______________________________                                     \ /        _______________________________                                             _______________________________                                     \
|     4 /                2         2        /atan2(-28*im(c), 25 - 28*re(c))\     4 /                2         2        /atan2(-28*im(c), 25 - 28*re(c))\| |     4 /                2         2        /atan2(-28*im(c), 25 - 28*re(c))\     4 /                2         2        /atan2(-28*im(c), 25 - 28*re(c))\|
|     \/  (25 - 28*re(c))  + 784*im (c) *cos|-------------------------------|   I*\/  (25 - 28*re(c))  + 784*im (c) *sin|-------------------------------|| |     \/  (25 - 28*re(c))  + 784*im (c) *cos|-------------------------------|   I*\/  (25 - 28*re(c))  + 784*im (c) *sin|-------------------------------||
|5                                          \               2               /                                           \               2               /| |5                                          \               2               /                                           \               2               /|
|-- - ----------------------------------------------------------------------- - -------------------------------------------------------------------------|*|-- + ----------------------------------------------------------------------- + -------------------------------------------------------------------------|
\14                                      14                                                                         14                                   / \14                                      14                                                                         14                                   /
(−i(25−28re⁡(c))2+784(im⁡(c))24sin⁡(atan2⁡(−28im⁡(c),25−28re⁡(c))2)14−(25−28re⁡(c))2+784(im⁡(c))24cos⁡(atan2⁡(−28im⁡(c),25−28re⁡(c))2)14+514)(i(25−28re⁡(c))2+784(im⁡(c))24sin⁡(atan2⁡(−28im⁡(c),25−28re⁡(c))2)14+(25−28re⁡(c))2+784(im⁡(c))24cos⁡(atan2⁡(−28im⁡(c),25−28re⁡(c))2)14+514)\left(- \frac{i \sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{5}{14}\right) \left(\frac{i \sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{5}{14}\right) 
=
re(c)   I*im(c)
----- + -------
  7        7
re⁡(c)7+iim⁡(c)7\frac{\operatorname{re}{\left(c\right)}}{7} + \frac{i \operatorname{im}{\left(c\right)}}{7} 
re(c)/7 + i*im(c)/7
    © Mister Exam – Calculator