Mister Exam
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7x^2-5x+c=0 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
$$a = 7$$
$$b = -5$$
, then
The equation has two roots.
or
$$x_{1} = \frac{\sqrt{25 - 28 c}}{14} + \frac{5}{14}$$
$$x_{2} = \frac{5}{14} - \frac{\sqrt{25 - 28 c}}{14}$$
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
$$a = 7$$
$$b = -5$$
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
$$x_{1} = \frac{\sqrt{25 - 28 c}}{14} + \frac{5}{14}$$
$$x_{2} = \frac{5}{14} - \frac{\sqrt{25 - 28 c}}{14}$$
Vieta's Theorem
rewrite the equation
$$c + \left(7 x^{2} - 5 x\right) = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{c}{7} + x^{2} - \frac{5 x}{7} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{5}{7}$$
$$q = \frac{c}{a}$$
$$q = \frac{c}{7}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{5}{7}$$
$$x_{1} x_{2} = \frac{c}{7}$$
$$c + \left(7 x^{2} - 5 x\right) = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{c}{7} + x^{2} - \frac{5 x}{7} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{5}{7}$$
$$q = \frac{c}{a}$$
$$q = \frac{c}{7}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{5}{7}$$
$$x_{1} x_{2} = \frac{c}{7}$$
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
$$x_{1} = - \frac{i \sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{5}{14}$$
_______________________________ _______________________________
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
$$x_{2} = \frac{i \sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{5}{14}$$
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
$$\left(- \frac{i \sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{5}{14}\right) + \left(\frac{i \sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{5}{14}\right)$$
=
5/7
$$\frac{5}{7}$$
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 /
$$\left(- \frac{i \sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} - \frac{\sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{5}{14}\right) \left(\frac{i \sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{\sqrt[4]{\left(25 - 28 \operatorname{re}{\left(c\right)}\right)^{2} + 784 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{im}{\left(c\right)},25 - 28 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{14} + \frac{5}{14}\right)$$
=
re(c) I*im(c) ----- + ------- 7 7
$$\frac{\operatorname{re}{\left(c\right)}}{7} + \frac{i \operatorname{im}{\left(c\right)}}{7}$$
re(c)/7 + i*im(c)/7