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(6x+2)²=(6x-1)(5x+1) equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
2 (6*x + 2) = (6*x - 1)*(5*x + 1)
$$\left(6 x + 2\right)^{2} = \left(5 x + 1\right) \left(6 x - 1\right)$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$\left(6 x + 2\right)^{2} = \left(5 x + 1\right) \left(6 x - 1\right)$$
to
$$- \left(5 x + 1\right) \left(6 x - 1\right) + \left(6 x + 2\right)^{2} = 0$$
Expand the expression in the equation
$$- \left(5 x + 1\right) \left(6 x - 1\right) + \left(6 x + 2\right)^{2} = 0$$
We get the quadratic equation
$$6 x^{2} + 23 x + 5 = 0$$
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
$$a = 6$$
$$b = 23$$
$$c = 5$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = - \frac{23}{12} + \frac{\sqrt{409}}{12}$$
$$x_{2} = - \frac{23}{12} - \frac{\sqrt{409}}{12}$$
left part with negative sign.
The equation is transformed from
$$\left(6 x + 2\right)^{2} = \left(5 x + 1\right) \left(6 x - 1\right)$$
to
$$- \left(5 x + 1\right) \left(6 x - 1\right) + \left(6 x + 2\right)^{2} = 0$$
Expand the expression in the equation
$$- \left(5 x + 1\right) \left(6 x - 1\right) + \left(6 x + 2\right)^{2} = 0$$
We get the quadratic equation
$$6 x^{2} + 23 x + 5 = 0$$
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
$$a = 6$$
$$b = 23$$
$$c = 5$$
, then
D = b^2 - 4 * a * c =
(23)^2 - 4 * (6) * (5) = 409
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{23}{12} + \frac{\sqrt{409}}{12}$$
$$x_{2} = - \frac{23}{12} - \frac{\sqrt{409}}{12}$$
Rapid solution
[src]
_____
23 \/ 409
x1 = - -- - -------
12 12
$$x_{1} = - \frac{23}{12} - \frac{\sqrt{409}}{12}$$
_____
23 \/ 409
x2 = - -- + -------
12 12
$$x_{2} = - \frac{23}{12} + \frac{\sqrt{409}}{12}$$
x2 = -23/12 + sqrt(409)/12
Sum and product of roots
[src]
sum
_____ _____ 23 \/ 409 23 \/ 409 - -- - ------- + - -- + ------- 12 12 12 12
$$\left(- \frac{23}{12} - \frac{\sqrt{409}}{12}\right) + \left(- \frac{23}{12} + \frac{\sqrt{409}}{12}\right)$$
=
-23/6
$$- \frac{23}{6}$$
product
/ _____\ / _____\ | 23 \/ 409 | | 23 \/ 409 | |- -- - -------|*|- -- + -------| \ 12 12 / \ 12 12 /
$$\left(- \frac{23}{12} - \frac{\sqrt{409}}{12}\right) \left(- \frac{23}{12} + \frac{\sqrt{409}}{12}\right)$$
=
5/6
$$\frac{5}{6}$$
5/6