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((2x+1)^2)/25-(x-1)/3=x equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
2
(2*x + 1) x - 1
---------- - ----- = x
25 3
$$- \frac{x - 1}{3} + \frac{\left(2 x + 1\right)^{2}}{25} = x$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$- \frac{x - 1}{3} + \frac{\left(2 x + 1\right)^{2}}{25} = x$$
to
$$- x + \left(- \frac{x - 1}{3} + \frac{\left(2 x + 1\right)^{2}}{25}\right) = 0$$
Expand the expression in the equation
$$- x + \left(- \frac{x - 1}{3} + \frac{\left(2 x + 1\right)^{2}}{25}\right) = 0$$
We get the quadratic equation
$$\frac{4 x^{2}}{25} - \frac{88 x}{75} + \frac{28}{75} = 0$$
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
$$a = \frac{4}{25}$$
$$b = - \frac{88}{75}$$
$$c = \frac{28}{75}$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 7$$
$$x_{2} = \frac{1}{3}$$
left part with negative sign.
The equation is transformed from
$$- \frac{x - 1}{3} + \frac{\left(2 x + 1\right)^{2}}{25} = x$$
to
$$- x + \left(- \frac{x - 1}{3} + \frac{\left(2 x + 1\right)^{2}}{25}\right) = 0$$
Expand the expression in the equation
$$- x + \left(- \frac{x - 1}{3} + \frac{\left(2 x + 1\right)^{2}}{25}\right) = 0$$
We get the quadratic equation
$$\frac{4 x^{2}}{25} - \frac{88 x}{75} + \frac{28}{75} = 0$$
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
$$a = \frac{4}{25}$$
$$b = - \frac{88}{75}$$
$$c = \frac{28}{75}$$
, then
D = b^2 - 4 * a * c =
(-88/75)^2 - 4 * (4/25) * (28/75) = 256/225
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 7$$
$$x_{2} = \frac{1}{3}$$
Sum and product of roots
[src]
sum
7 + 1/3
$$\frac{1}{3} + 7$$
=
22/3
$$\frac{22}{3}$$
product
7 - 3
$$\frac{7}{3}$$
=
7/3
$$\frac{7}{3}$$
7/3