Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x/4+x=4
-
Equation x^2+11=0
- Equation x+y=5
-
Equation 1*(x-1)*(x-3)=0
- Express {x} in terms of y where:
- -11*x-3*y=14
- -1*x-17*y=-12
- 20*x+18*y=-9
- -2*x+20*y=-2
-
Identical expressions
- (- five *x- three)*(two *x- one)-a= zero
- ( minus 5 multiply by x minus 3) multiply by (2 multiply by x minus 1) minus a equally 0
- ( minus five multiply by x minus three) multiply by (two multiply by x minus one) minus a equally zero
- (-5x-3)(2x-1)-a=0
- -5x-32x-1-a=0
- (-5*x-3)*(2*x-1)-a=O
-
Similar expressions
- (-5*x-3)*(2*x-1)+a=0
- (-5*x+3)*(2*x-1)-a=0
- (-5*x-3)*(2*x+1)-a=0
- (5*x-3)*(2*x-1)-a=0
(-5*x-3)*(2*x-1)-a=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
(-5*x - 3)*(2*x - 1) - a = 0
$$- a + \left(- 5 x - 3\right) \left(2 x - 1\right) = 0$$
Detail solution
Expand the expression in the equation
$$- a + \left(- 5 x - 3\right) \left(2 x - 1\right) = 0$$
We get the quadratic equation
$$- a - 10 x^{2} - x + 3 = 0$$
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
$$a = -10$$
$$b = -1$$
$$c = 3 - a$$
, then
The equation has two roots.
or
$$x_{1} = - \frac{\sqrt{121 - 40 a}}{20} - \frac{1}{20}$$
$$x_{2} = \frac{\sqrt{121 - 40 a}}{20} - \frac{1}{20}$$
$$- a + \left(- 5 x - 3\right) \left(2 x - 1\right) = 0$$
We get the quadratic equation
$$- a - 10 x^{2} - x + 3 = 0$$
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
$$a = -10$$
$$b = -1$$
$$c = 3 - a$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (-10) * (3 - a) = 121 - 40*a
The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{\sqrt{121 - 40 a}}{20} - \frac{1}{20}$$
$$x_{2} = \frac{\sqrt{121 - 40 a}}{20} - \frac{1}{20}$$
The graph
Sum and product of roots
[src]
sum
_________________________________ _________________________________ _________________________________ _________________________________
4 / 2 2 /atan2(-40*im(a), 121 - 40*re(a))\ 4 / 2 2 /atan2(-40*im(a), 121 - 40*re(a))\ 4 / 2 2 /atan2(-40*im(a), 121 - 40*re(a))\ 4 / 2 2 /atan2(-40*im(a), 121 - 40*re(a))\
\/ (121 - 40*re(a)) + 1600*im (a) *cos|--------------------------------| I*\/ (121 - 40*re(a)) + 1600*im (a) *sin|--------------------------------| \/ (121 - 40*re(a)) + 1600*im (a) *cos|--------------------------------| I*\/ (121 - 40*re(a)) + 1600*im (a) *sin|--------------------------------|
1 \ 2 / \ 2 / 1 \ 2 / \ 2 /
- -- - -------------------------------------------------------------------------- - ---------------------------------------------------------------------------- + - -- + -------------------------------------------------------------------------- + ----------------------------------------------------------------------------
20 20 20 20 20 20
$$\left(- \frac{i \sqrt[4]{\left(121 - 40 \operatorname{re}{\left(a\right)}\right)^{2} + 1600 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 40 \operatorname{im}{\left(a\right)},121 - 40 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{20} - \frac{\sqrt[4]{\left(121 - 40 \operatorname{re}{\left(a\right)}\right)^{2} + 1600 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 40 \operatorname{im}{\left(a\right)},121 - 40 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{20} - \frac{1}{20}\right) + \left(\frac{i \sqrt[4]{\left(121 - 40 \operatorname{re}{\left(a\right)}\right)^{2} + 1600 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 40 \operatorname{im}{\left(a\right)},121 - 40 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{20} + \frac{\sqrt[4]{\left(121 - 40 \operatorname{re}{\left(a\right)}\right)^{2} + 1600 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 40 \operatorname{im}{\left(a\right)},121 - 40 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{20} - \frac{1}{20}\right)$$
=
-1/10
$$- \frac{1}{10}$$
product
/ _________________________________ _________________________________ \ / _________________________________ _________________________________ \ | 4 / 2 2 /atan2(-40*im(a), 121 - 40*re(a))\ 4 / 2 2 /atan2(-40*im(a), 121 - 40*re(a))\| | 4 / 2 2 /atan2(-40*im(a), 121 - 40*re(a))\ 4 / 2 2 /atan2(-40*im(a), 121 - 40*re(a))\| | \/ (121 - 40*re(a)) + 1600*im (a) *cos|--------------------------------| I*\/ (121 - 40*re(a)) + 1600*im (a) *sin|--------------------------------|| | \/ (121 - 40*re(a)) + 1600*im (a) *cos|--------------------------------| I*\/ (121 - 40*re(a)) + 1600*im (a) *sin|--------------------------------|| | 1 \ 2 / \ 2 /| | 1 \ 2 / \ 2 /| |- -- - -------------------------------------------------------------------------- - ----------------------------------------------------------------------------|*|- -- + -------------------------------------------------------------------------- + ----------------------------------------------------------------------------| \ 20 20 20 / \ 20 20 20 /
$$\left(- \frac{i \sqrt[4]{\left(121 - 40 \operatorname{re}{\left(a\right)}\right)^{2} + 1600 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 40 \operatorname{im}{\left(a\right)},121 - 40 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{20} - \frac{\sqrt[4]{\left(121 - 40 \operatorname{re}{\left(a\right)}\right)^{2} + 1600 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 40 \operatorname{im}{\left(a\right)},121 - 40 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{20} - \frac{1}{20}\right) \left(\frac{i \sqrt[4]{\left(121 - 40 \operatorname{re}{\left(a\right)}\right)^{2} + 1600 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 40 \operatorname{im}{\left(a\right)},121 - 40 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{20} + \frac{\sqrt[4]{\left(121 - 40 \operatorname{re}{\left(a\right)}\right)^{2} + 1600 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 40 \operatorname{im}{\left(a\right)},121 - 40 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{20} - \frac{1}{20}\right)$$
=
3 re(a) I*im(a) - -- + ----- + ------- 10 10 10
$$\frac{\operatorname{re}{\left(a\right)}}{10} + \frac{i \operatorname{im}{\left(a\right)}}{10} - \frac{3}{10}$$
-3/10 + re(a)/10 + i*im(a)/10
Rapid solution
[src]
_________________________________ _________________________________
4 / 2 2 /atan2(-40*im(a), 121 - 40*re(a))\ 4 / 2 2 /atan2(-40*im(a), 121 - 40*re(a))\
\/ (121 - 40*re(a)) + 1600*im (a) *cos|--------------------------------| I*\/ (121 - 40*re(a)) + 1600*im (a) *sin|--------------------------------|
1 \ 2 / \ 2 /
x1 = - -- - -------------------------------------------------------------------------- - ----------------------------------------------------------------------------
20 20 20
$$x_{1} = - \frac{i \sqrt[4]{\left(121 - 40 \operatorname{re}{\left(a\right)}\right)^{2} + 1600 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 40 \operatorname{im}{\left(a\right)},121 - 40 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{20} - \frac{\sqrt[4]{\left(121 - 40 \operatorname{re}{\left(a\right)}\right)^{2} + 1600 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 40 \operatorname{im}{\left(a\right)},121 - 40 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{20} - \frac{1}{20}$$
_________________________________ _________________________________
4 / 2 2 /atan2(-40*im(a), 121 - 40*re(a))\ 4 / 2 2 /atan2(-40*im(a), 121 - 40*re(a))\
\/ (121 - 40*re(a)) + 1600*im (a) *cos|--------------------------------| I*\/ (121 - 40*re(a)) + 1600*im (a) *sin|--------------------------------|
1 \ 2 / \ 2 /
x2 = - -- + -------------------------------------------------------------------------- + ----------------------------------------------------------------------------
20 20 20
$$x_{2} = \frac{i \sqrt[4]{\left(121 - 40 \operatorname{re}{\left(a\right)}\right)^{2} + 1600 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 40 \operatorname{im}{\left(a\right)},121 - 40 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{20} + \frac{\sqrt[4]{\left(121 - 40 \operatorname{re}{\left(a\right)}\right)^{2} + 1600 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 40 \operatorname{im}{\left(a\right)},121 - 40 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{20} - \frac{1}{20}$$
x2 = i*((121 - 40*re(a))^2 + 1600*im(a)^2)^(1/4)*sin(atan2(-40*im(a, 121 - 40*re(a))/2)/20 + ((121 - 40*re(a))^2 + 1600*im(a)^2)^(1/4)*cos(atan2(-40*im(a), 121 - 40*re(a))/2)/20 - 1/20)